# OEIS Recent Additions (http://oeis.org/recent.txt) # Last Modified: February 7 09:57 UTC 2018 # Use of this content is governed by the # OEIS End-User License: http://oeis.org/LICENSE %I A299332 %S A299332 0,41,858,22718,816886,27685946,967671172,33782479865,1181914438937, %T A299332 41389407999970,1449682526889422,50785750873434402, %U A299332 1779232817481119797,62335724066481284310,2183970192528307684067,76517142801168870666049 %N A299332 Number of nX6 0..1 arrays with every element equal to 2, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299332 Column 6 of A299334. %H A299332 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299332 Some solutions for n=5 %e A299332 ..0..1..1..1..0..0. .0..0..1..1..0..0. .0..1..1..0..0..0. .0..0..0..1..1..1 %e A299332 ..0..0..1..0..0..1. .0..0..1..1..1..0. .0..0..1..0..1..0. .0..0..0..1..1..0 %e A299332 ..0..0..0..0..1..1. .0..0..1..1..1..1. .0..0..1..1..1..1. .0..0..1..0..0..0 %e A299332 ..0..0..1..0..0..0. .0..1..1..1..0..1. .0..1..0..1..0..1. .0..0..1..0..0..1 %e A299332 ..0..0..1..1..0..0. .0..0..0..0..0..0. .0..0..1..0..0..0. .0..1..1..1..1..1 %Y A299332 Cf. A299334. %K A299332 nonn,new %O A299332 1,2 %A A299332 _R. H. Hardin_, Feb 07 2018 %I A299331 %S A299331 0,17,173,2537,46286,816886,14783424,267652693,4856785087,88215895898, %T A299331 1602696122814,29123485135149,529251045420760,9618213506625893, %U A299331 174796707221176201,3176688870307290172,57732099267860132387 %N A299331 Number of nX5 0..1 arrays with every element equal to 2, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299331 Column 5 of A299334. %H A299331 R. H. Hardin, Table of n, a(n) for n = 1..210 %H A299331 R. H. Hardin, Empirical recurrence of order 76 %F A299331 Empirical recurrence of order 76 (see link above) %e A299331 Some solutions for n=5 %e A299331 ..0..0..0..1..1. .0..1..1..1..1. .0..1..1..0..0. .0..1..1..1..0 %e A299331 ..0..0..1..1..0. .0..0..1..1..0. .0..0..1..1..0. .0..0..1..0..0 %e A299331 ..0..1..1..0..0. .1..0..1..0..0. .0..1..0..1..0. .0..0..1..0..1 %e A299331 ..1..1..1..1..0. .1..1..0..1..1. .0..0..1..0..1. .1..0..0..1..1 %e A299331 ..1..1..1..1..1. .1..0..0..0..1. .0..1..1..1..1. .1..1..1..1..1 %Y A299331 Cf. A299334. %K A299331 nonn,new %O A299331 1,2 %A A299331 _R. H. Hardin_, Feb 07 2018 %I A299330 %S A299330 0,6,36,263,2537,22718,214683,2024559,19169227,181762287,1723995924, %T A299330 16356826774,155199770911,1472657804158,13973960703276, %U A299330 132598941011278,1258235426682140,11939448572555797,113293986222690113 %N A299330 Number of nX4 0..1 arrays with every element equal to 2, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299330 Column 4 of A299334. %H A299330 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299330 Empirical: a(n) = 7*a(n-1) +42*a(n-2) -97*a(n-3) -760*a(n-4) -293*a(n-5) +4331*a(n-6) +7298*a(n-7) -1378*a(n-8) -15896*a(n-9) -25556*a(n-10) -16583*a(n-11) +646*a(n-12) +14623*a(n-13) +22835*a(n-14) +24608*a(n-15) +8564*a(n-16) -17982*a(n-17) +3869*a(n-18) -524*a(n-19) -11907*a(n-20) -6997*a(n-21) +8341*a(n-22) +3006*a(n-23) -7404*a(n-24) +2016*a(n-25) +1712*a(n-26) -680*a(n-27) for n>28 %e A299330 Some solutions for n=5 %e A299330 ..0..0..0..1. .0..0..0..0. .0..0..1..1. .0..0..1..1. .0..0..1..1 %e A299330 ..0..0..1..1. .1..0..0..0. .0..0..0..1. .0..1..1..1. .1..0..1..1 %e A299330 ..0..1..1..1. .1..1..1..0. .0..0..0..1. .0..0..0..1. .1..1..0..0 %e A299330 ..1..1..0..0. .0..1..0..1. .1..0..1..0. .1..1..0..0. .0..1..0..1 %e A299330 ..1..1..0..0. .0..0..1..1. .1..1..0..0. .1..0..0..0. .0..0..1..1 %Y A299330 Cf. A299334. %K A299330 nonn,new %O A299330 1,2 %A A299330 _R. H. Hardin_, Feb 07 2018 %I A299329 %S A299329 0,3,8,36,173,858,4258,21386,107465,541047,2725498,13733762,69215100, %T A299329 348852964,1758324488,8862635135,44671460074,225164071583, %U A299329 1134929377517,5720565544510,28834291127949,145338167739511,732571679461791 %N A299329 Number of nX3 0..1 arrays with every element equal to 2, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299329 Column 3 of A299334. %H A299329 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299329 Empirical: a(n) = 4*a(n-1) +10*a(n-2) -13*a(n-3) -53*a(n-4) -25*a(n-5) +57*a(n-6) +53*a(n-7) +a(n-8) +16*a(n-9) +10*a(n-10) -15*a(n-11) -4*a(n-12) for n>14 %e A299329 Some solutions for n=5 %e A299329 ..0..1..1. .0..1..1. .0..0..0. .0..0..1. .0..0..0. .0..0..0. .0..1..1 %e A299329 ..0..0..1. .0..0..1. .0..0..1. .0..1..1. .0..0..0. .0..1..0. .0..0..1 %e A299329 ..0..1..1. .0..0..0. .0..1..1. .1..1..1. .0..1..0. .1..0..1. .1..1..0 %e A299329 ..0..1..1. .0..1..0. .0..1..0. .1..0..0. .0..1..1. .1..1..0. .0..1..0 %e A299329 ..0..0..1. .1..1..1. .0..0..0. .0..0..0. .0..0..1. .1..0..0. .0..0..0 %Y A299329 Cf. A299334. %K A299329 nonn,new %O A299329 1,2 %A A299329 _R. H. Hardin_, Feb 07 2018 %I A299328 %S A299328 0,1,8,263,46286,27685946,65181402152,567774280392040, %T A299328 18419330490239501061 %N A299328 Number of nXn 0..1 arrays with every element equal to 2, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299328 Diagonal of A299334. %e A299328 Some solutions for n=5 %e A299328 ..0..0..0..1..1. .0..0..0..0..0. .0..0..1..1..0. .0..0..0..1..1 %e A299328 ..0..1..1..0..1. .0..0..0..1..0. .0..1..1..0..0. .0..0..1..0..1 %e A299328 ..1..0..1..0..1. .1..1..1..0..1. .1..1..1..0..0. .1..0..1..1..0 %e A299328 ..1..0..0..1..1. .0..1..0..1..1. .1..1..0..0..0. .1..1..1..1..0 %e A299328 ..1..1..0..0..1. .0..0..0..1..1. .1..0..0..0..0. .1..1..1..0..0 %Y A299328 Cf. A299334. %K A299328 nonn,new %O A299328 1,3 %A A299328 _R. H. Hardin_, Feb 07 2018 %I A298485 %S A298485 1,2,-1,2,1,-1,2,3,0,-1,2,5,3,-1,-1,2,7,8,2,-2,-1,2,9,15,10,0,-3,-1,2, %T A298485 11,24,25,10,-3,-4,-1,2,13,35,49,35,7,-7,-5,-1,2,15,48,84,84,42,0,-12, %U A298485 -6,-1,2,17,63,132,168,126,42,-12,-18,-7,-1,2,19,80,195,300,294,168,30,-30,-25,-8,-1 %N A298485 Triangle read by rows; row 0 is 1; the n-th row for n>0 contains the coefficients in the expansion of (2-x)*(1+x)^(n-1). %F A298485 T(n,k) = T(n-1,k-1)+T(n-1,k); T(0,0)=1, T(1,0)=2, T(1,1)=-1. %e A298485 Triangle begins: %e A298485 1; %e A298485 2, -1; %e A298485 2, 1, -1; %e A298485 2, 3, 0, -1; %e A298485 2, 5, 3, -1, -1; %e A298485 2, 7, 8, 2, -2, -1; %e A298485 2, 9, 15, 10, 0, -3, -1; %e A298485 2, 11, 24, 25, 10, -3, -4, -1; %e A298485 2, 13, 35, 49, 35, 7, -7, -5, -1; %e A298485 ... %t A298485 T[0, 0] = 1; T[_, 0] = 2; T[1, 1] = -1; T[n_?Positive, k_?Positive] := T[n, k] = T[n - 1, k - 1] + T[n - 1, k]; T[_, _] = 0; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 05 2018 *) %Y A298485 Central column gives A088218. %Y A298485 Cf. A037012. %K A298485 sign,tabl,new %O A298485 0,2 %A A298485 _Seiichi Manyama_, Jan 20 2018 %I A299205 %S A299205 2,3,10,14,15,56,57,59,70,85,105,107,116,136,209,267,295,323,352,393, %T A299205 415,442,530,551,645,646,760,855,1197,1288,1342,1415,1472,1496,1625, %U A299205 1765,1953,2002,2255,2485,2847,2945,3039,3382,3591,3745,3905,4233,4264,4313 %N A299205 Numbers k such that k-1 divides tau(k), where tau(k)=A000594(k) is Ramanujan's tau function. %C A299205 Numbers k such that A299204(k) = 0. %H A299205 Eric Weisstein's World of Mathematics, Tau Function %o A299205 (PARI) isok(n) = (ramanujantau(n) % (n-1)) == 0; \\ _Michel Marcus_, Feb 05 2018 %Y A299205 For the sequence when n is prime see A299172. %Y A299205 Cf. A000594, A063938, A299157, A299204. %K A299205 nonn,new %O A299205 1,1 %A A299205 _Seiichi Manyama_, Feb 05 2018 %I A298044 %S A298044 15,15,30,45,10,63,99,55,52,187,195,374,33,396,418,286,510,570,572, %T A298044 490,221,63,273,910,1050,1092,660,1160,882,885,1148,1485,1505,1092, %U A298044 495,1870,308,858,1485,585,990,385,1333,1001,2907,3045,2277,1394,1475,2310,3690,594,3885,3465,2755,1496,4422,2730,3393 %N A298044 1/16 of the edge with the largest 2-adic valuation of a primitive 3-simplex (0, b=A031173, c=A031174, d=A031175). %C A298044 Each primitive 3-simplex (0,b,c,d) with b > c > d is built by an odd, an even and an edge with the largest 2-adic valuation. %H A298044 Eric Weisstein's World of Mathematics, Trirectangular Tetrahedron %Y A298044 Cf. A298046, A298047, A031173, A031174, A031175. %K A298044 nonn,new %O A298044 1,1 %A A298044 _Ralf Steiner_, Jan 11 2018 %I A298046 %S A298046 11,63,35,33,198,275,255,585,660,195,207,390,1449,187,1575,1683,1222, %T A298046 418,741,2457,2805,275,3465,270,2574,2205,4437,1518,3850,5369,5940, %U A298046 1225,6171,6426,3808,1950,7695,1890,8901,8976,9275,741,6435,297,10395,4615,12831,12870,13299,2133,13570,7843,10593,4901 %N A298046 1/4 of the even edge of least 2-adic valuation of a primitive 3-simplex (0, b=A031173, c=A031174, d=A031175). %C A298046 Each primitive 3-simplex (0,b,c,d) with b > c > d is built by 1 odd edge and 2 even edges; the edge that is considered here is the even one with the least 2-adic valuation. %H A298046 Eric Weisstein's World of Mathematics, Trirectangular Tetrahedron %Y A298046 Cf. A298044, A298047, A031173, A031174, A031175. %K A298046 nonn,new %O A298046 1,1 %A A298046 _Ralf Steiner_, Jan 11 2018 %I A298047 %S A298047 117,275,693,85,231,1155,187,429,855,2475,2035,2295,6325,195,1155, %T A298047 1755,495,1575,9405,10725,2925,12075,4901,1881,11753,7579,8789,16929, %U A298047 19305,5643,4599,17157,6435,935,26649,23751,10725,35321,35075,1105,2163,38475,38571,39195,5491,15939,51205,24225,9405,57275 %N A298047 The odd edge of a primitive 3-simplex (0, b=A031173, c=A031174, d=A031175). %C A298047 Each primitive 3-simplex (0,b,c,d) with b > c > d is built by 1 odd edge and 2 even edges; the edge that is considered here is the odd edge. %H A298047 Eric Weisstein's World of Mathematics, Trirectangular Tetrahedron %Y A298047 Cf. A298044, A298046, A031173, A031174, A031175. %K A298047 nonn,new %O A298047 1,1 %A A298047 _Ralf Steiner_, Jan 11 2018 %I A299254 %S A299254 1,7,21,45,79,122,175,237,309,391,482,583,693,813,943,1082,1231,1389, %T A299254 1557,1735,1922,2119,2325,2541,2767,3002,3247,3501,3765,4039,4322, %U A299254 4615,4917,5229,5551,5882,6223,6573,6933,7303,7682,8071,8469,8877,9295,9722,10159,10605,11061,11527,12002 %N A299254 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3^4.6 2D tiling (cf. A250120). %F A299254 G.f.: (x^2+x+1)*(x^4+3*x^3+3*x+1)*(x+1) / ((x^4+x^3+x^2+x+1)*(1-x)^3). %Y A299254 Cf. A250120. %K A299254 nonn,new %O A299254 0,2 %A A299254 _N. J. A. Sloane_, Feb 06 2018 %I A299156 %S A299156 1,256,397,1197,8053,8736,9901,32173,33493,33757,38461,48757,56101, %T A299156 57073,64153,76561,79693,87517,100608,102217,105253,105601,105913, %U A299156 105997,107713,108553,110976,116293,123121,131437,138517,143137,147541,151237,156601,171253 %N A299156 Numbers k such that k*(k+1) divides tribonacci(k) (A000073(k)). %C A299156 A subsequence of A232570. %e A299156 tribonacci(256) = 10285895715599251294835119279496333059462348558276025598603904254464 = 256 * 257 * 156339611436029476149609668037091638184921397104146789862048642. %p A299156 with(LinearAlgebra[Modular]): %p A299156 T:= (n, m)-> MatrixPower(m, Mod(m, <<0|1|0>, %p A299156 <0|0|1>, <1|1|1>>, float[8]), n)[1, 3]: %p A299156 a:= proc(n) option remember; local i, k, ok; %p A299156 if n=1 then 1 else %p A299156 for k from 1+a(n-1) do ok:= true; %p A299156 for i in ifactors(k*(k+1)/2)[2] while ok do %p A299156 ok:= is(T(k, i[1]^i[2])=0) %p A299156 od; if ok then break fi %p A299156 od; k %p A299156 fi %p A299156 end: %p A299156 seq(a(n), n=1..10); # _Alois P. Heinz_, Feb 06 2018 %Y A299156 Cf. A000073, A217738, A232570, A274518. %K A299156 nonn,new %O A299156 1,2 %A A299156 _Seiichi Manyama_, Feb 04 2018 %I A298000 %S A298000 1,2,10,13,16,19,22,27,29,34,36,41,43,48,50,55,57,60,63,68,72,74,77, %T A298000 80,85,89,91,94,97,102,106,108,111,114,119,123,125,128,131,136,140, %U A298000 142,147,149,154,156,159,162,167,169,172,177,181,183,188,190,195,197 %N A298000 Solution of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments. %C A298000 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. %C A298000 Conjectures: a(n) - (2 +sqrt(2))*n < 4 for n >= 1. Guide to related sequences having initial values a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, where (b(n)) is the increasing sequence of positive integers not in (a(n)): %C A298000 *** %C A298000 a(n) = a(1)*b(n) - a(0)*b(n-1) + n (a(n)) = A297999; (b(n)) = A298110 %C A298000 a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*n (a(n)) = A298000; (b(n)) = A298111 %C A298000 a(n) = a(1)*b(n) - a(0)*b(n-1) + 3*n (a(n)) = A298001; (b(n)) = A298112 %C A298000 a(n) = a(1)*b(n) - a(0)*b(n-1) + 4*n (a(n)) = A298002; (b(n)) = A298113 %H A298000 Clark Kimberling, Table of n, a(n) for n = 0..10000 %e A298000 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 10. %e A298000 Complement: (b(n)) = (3,4,5,6,8,9,11,12,14,15,17,18,20,...) %t A298000 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5; %t A298000 a[n_] := a[n] = a[1]*b[n] - a[0]*b[n - 1] + 2 n; %t A298000 j = 1; While[j < 100, k = a[j] - j - 1; %t A298000 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k %t A298000 Table[a[n], {n, 0, k}] (* A298000 *) %Y A298000 Cf. A297826, A297836, A297837. %K A298000 nonn,easy,new %O A298000 0,2 %A A298000 _Clark Kimberling_, Feb 04 2018 %I A297998 %S A297998 1,2,10,13,17,20,24,27,33,35,41,43,47,52,55,60,63,66,72,74,80,82,86, %T A297998 89,93,98,103,105,109,112,116,121,126,128,132,137,140,143,147,152,155, %U A297998 160,163,166,170,175,178,183,186,191,194,197,201,204,210,214,217 %N A297998 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + floor(5*n/2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments. %C A297998 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. %H A297998 Clark Kimberling, Table of n, a(n) for n = 0..10000 %e A297998 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 10. %e A297998 Complement: (b(n)) = (3,4,5,6,7,8,9,11,12,14,15,16,18,19,21,...) %t A297998 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; %t A297998 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + Floor[5/2]; %t A297998 j = 1; While[j < 100, k = a[j] - j - 1; %t A297998 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k %t A297998 Table[a[n], {n, 0, k}] (* A297998 *) %Y A297998 Cf. A297826, A297830, A297836. %K A297998 nonn,easy,new %O A297998 0,2 %A A297998 _Clark Kimberling_, Feb 04 2018 %I A297837 %S A297837 1,2,13,18,23,28,33,38,43,48,53,60,64,69,74,81,85,90,95,102,106,111, %T A297837 116,123,127,132,137,144,148,153,158,165,169,174,179,186,190,195,200, %U A297837 207,211,216,221,228,232,237,242,247,252,259,263,268,275,279,284,289 %N A297837 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 4*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments. %C A297837 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. For a guide to related sequences, see A297830. %C A297837 Conjecture: a(n) - (3 + sqrt(5))*n < 3 for n >= 1. %H A297837 Clark Kimberling, Table of n, a(n) for n = 0..10000 %e A297837 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 13. %e A297837 Complement: (b(n)) = (3,4,5,6,7,8,9,10,11,12,14,15,16,17,19,20,...) %t A297837 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; %t A297837 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 4 n; %t A297837 j = 1; While[j < 100, k = a[j] - j - 1; %t A297837 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k %t A297837 Table[a[n], {n, 0, k}] (* A297836 *) %Y A297837 Cf. A297826, A297830, A297836. %K A297837 nonn,easy,new %O A297837 0,2 %A A297837 _Clark Kimberling_, Feb 04 2018 %I A297836 %S A297836 1,2,11,15,19,23,27,31,35,41,44,48,54,57,61,67,70,74,80,83,87,93,96, %T A297836 100,106,109,113,119,122,126,130,134,140,143,149,152,156,162,165,169, %U A297836 173,177,183,186,192,195,199,205,208,212,216,220,226,229,235,238,242 %N A297836 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 3*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments. %C A297836 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. For a guide to related sequences, see A297830. %C A297836 Conjectures: a(n) - (5 + sqrt(13))*n/2 < 2 for n >= 1. %H A297836 Clark Kimberling, Table of n, a(n) for n = 0..10000 %e A297836 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 11. %e A297836 Complement: (b(n)) = (3,4,5,6,7,8,9,10,12,13,14,16,17,18,20,...) %t A297836 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; %t A297836 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 3 n; %t A297836 j = 1; While[j < 100, k = a[j] - j - 1; %t A297836 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k %t A297836 Table[a[n], {n, 0, k}] (* A297836 *) %Y A297836 Cf. A297826, A297830, A297837. %K A297836 nonn,easy,new %O A297836 0,2 %A A297836 _Clark Kimberling_, Feb 04 2018 %I A297835 %S A297835 1,2,10,13,16,19,22,25,30,32,37,39,44,46,51,53,58,60,65,67,70,73,78, %T A297835 82,84,87,90,95,99,101,104,107,112,116,118,121,124,129,133,135,138, %U A297835 141,146,150,152,155,158,163,167,169,174,176,181,183,186,189,194,196 %N A297835 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n + 1, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments. %C A297835 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A297830 for a guide to related sequences. %C A297835 Conjecture: a(n) - (2 +sqrt(2))*n < 7 for n >= 1. %H A297835 Clark Kimberling, Table of n, a(n) for n = 0..10000 %e A297835 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 10. %e A297835 Complement: (b(n)) = (3,4,6,7,8,9,11,12,14,15,17,18,20,...) %t A297835 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; %t A297835 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 2 n + 1; %t A297835 j = 1; While[j < 100, k = a[j] - j - 1; %t A297835 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k %t A297835 Table[a[n], {n, 0, k}] (* A297835 *) %Y A297835 Cf. A297826, A297830. %K A297835 nonn,easy,new %O A297835 0,2 %A A297835 _Clark Kimberling_, Feb 04 2018 %I A297834 %S A297834 1,2,5,8,12,17,19,22,27,29,32,35,40,44,46,51,53,56,59,64,68,70,75,77, %T A297834 82,84,87,90,95,97,100,105,109,111,114,117,122,126,128,133,135,140, %U A297834 142,145,148,153,155,158,163,167,169,172,175,180,184,186,189,192 %N A297834 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 4, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments. %C A297834 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A297830 for a guide to related sequences. %C A297834 Conjecture: -3 < a(n) - (2 +sqrt(2))*n <= 1 for n >= 1. %H A297834 Clark Kimberling, Table of n, a(n) for n = 0..10000 %e A297834 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 5. %e A297834 Complement: (b(n)) = (3,4,6,7,9,10,11,13,14,15,16,18,20,...) %t A297834 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; %t A297834 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 2 n - 4; %t A297834 j = 1; While[j < 100, k = a[j] - j - 1; %t A297834 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k %t A297834 Table[a[n], {n, 0, k}] (* A297834 *) %Y A297834 Cf. A297826, A297830. %K A297834 nonn,easy,new %O A297834 0,2 %A A297834 _Clark Kimberling_, Feb 04 2018 %I A297833 %S A297833 1,2,6,9,14,16,21,23,26,29,34,38,40,43,46,51,55,57,62,64,69,71,74,77, %T A297833 82,84,87,92,96,98,103,105,110,112,115,118,123,125,128,133,137,139, %U A297833 142,145,150,154,156,159,162,167,171,173,178,180,185,187,190,193 %N A297833 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 3, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments. %C A297833 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A297830 for a guide to related sequences. %C A297833 Conjecture: -2 < a(n) - (2 +sqrt(2))*n <= 1 for n >= 1. %H A297833 Clark Kimberling, Table of n, a(n) for n = 0..10000 %e A297833 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 6. %e A297833 Complement: (b(n)) = (3,4,5,7,8,10,12,13,15,17,18,19,...) %t A297833 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; %t A297833 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 2 n - 3; %t A297833 j = 1; While[j < 100, k = a[j] - j - 1; %t A297833 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k %t A297833 Table[a[n], {n, 0, k}] (* A297833 *) %Y A297833 Cf. A297826, A297830. %K A297833 nonn,easy,new %O A297833 0,2 %A A297833 _Clark Kimberling_, Feb 04 2018 %I A297832 %S A297832 1,2,7,10,13,18,20,25,27,32,34,37,40,45,49,51,54,57,62,66,68,71,74,79, %T A297832 83,85,90,92,97,99,102,105,110,112,115,120,124,126,131,133,138,140, %U A297832 143,146,151,153,156,161,165,167,172,174,179,181,184,187,192,194 %N A297832 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 2, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments. %C A297832 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A297830 for a guide to related sequences. %C A297832 a(n) - (2+sqrt(2))*n < 2 for n >= 1. %H A297832 Clark Kimberling, Table of n, a(n) for n = 0..9999 %e A297832 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 7. %e A297832 Complement: (b(n)) = (3,4,5,7,8,10,12,13,15,17,18,19,...) %t A297832 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; %t A297832 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 2 n - 2; %t A297832 j = 1; While[j < 100, k = a[j] - j - 1; %t A297832 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k %t A297832 Table[a[n], {n, 0, k}] (* A297832 *) %Y A297832 Cf. A297826, A297830. %K A297832 nonn,easy,new %O A297832 0,2 %A A297832 _Clark Kimberling_, Feb 04 2018 %I A297831 %S A297831 1,2,8,11,14,17,22,24,29,31,36,38,43,45,48,51,56,60,62,65,68,73,77,79, %T A297831 82,85,90,94,96,99,102,107,111,113,118,120,125,127,130,133,138,140, %U A297831 143,148,152,154,159,161,166,168,171,174,179,181,184,189,193,195 %N A297831 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 1, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments. %C A297831 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A297830 for a guide to related sequences. %C A297831 Conjecture: a(n) - (2 +sqrt(2))*n < 5/2 for n >= 1. %H A297831 Clark Kimberling, Table of n, a(n) for n = 0..10000 %e A297831 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 8. %e A297831 Complement: (b(n)) = (3,4,5,6,7,9,10,12,13,15,16,18,19,...) %t A297831 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; %t A297831 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 2 n - 1; %t A297831 j = 1; While[j < 100, k = a[j] - j - 1; %t A297831 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k %t A297831 Table[a[n], {n, 0, k}] (* A297831 *) %Y A297831 Cf. A297826, A297830. %K A297831 nonn,easy,new %O A297831 0,2 %A A297831 _Clark Kimberling_, Feb 04 2018 %I A297830 %S A297830 1,2,9,12,15,18,21,26,28,33,35,40,42,47,49,54,56,59,62,67,71,73,76,79, %T A297830 84,88,90,93,96,101,105,107,110,113,118,122,124,127,130,135,139,141, %U A297830 146,148,153,155,158,161,166,168,171,176,180,182,187,189,194,196 %N A297830 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments. %C A297830 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. Conjecture: a(n) - (2 +sqrt(2))*n < 3 for n >= 1. %C A297830 Guide to related sequences having initial values a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, where (b(n)) is the increasing sequence of positive integers not in (a(n)): %C A297830 *** %C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + n (a(n)) = A297826; (b(n)) = A297997 %C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n (a(n)) = A297830; (b(n)) = A298003 %C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 3*n (a(n)) = A297836; (b(n)) = A298004 %C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 4*n (a(n)) = A297837; (b(n)) = A298005 %C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 1 (a(n)) = A297831; (b(n)) = A298006 %C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 2 (a(n)) = A297832; (b(n)) = A298007 %C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 3 (a(n)) = A297833; (b(n)) = A298108 %C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 4 (a(n)) = A297834; (b(n)) = A298209 %C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n + 1 (a(n)) = A297835; %C A297830 a(n) = a(1)*b(n-1) - a(0)*b(n-2)+floor(5*n/2) (a(n)) = A297998; %C A297830 *** %C A297830 For sequences (a(n)) and (b(n)) associated with equations of the form a(n) = a(1)*b(n) - a(0)*b(n-1), see the guide at A297800. %H A297830 Clark Kimberling, Table of n, a(n) for n = 0..10000 %e A297830 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 9. %e A297830 Complement: (b(n)) = (3,4,5,6,8,10,11,13,14,16,17,19,...) %t A297830 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; %t A297830 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 2 n; %t A297830 j = 1; While[j < 100, k = a[j] - j - 1; %t A297830 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k %t A297830 Table[a[n], {n, 0, k}] (* A297830 *) %Y A297830 Cf. A297826, A297836, A297837. %K A297830 nonn,easy,new %O A297830 0,2 %A A297830 _Clark Kimberling_, Feb 04 2018 %I A297829 %S A297829 51,59,243,279,287,295,363,371,422,430,538,587,684,872,934,1075,1083, %T A297829 1091,1232,1268,1304,1312,1320,1388,1396,1515,1598,1634,1642,1650, %U A297829 1718,1726,1855,1891,1899,1907,1975,1983,2034,2042,2093,2101,2209,2258,2355,2363 %N A297829 Duplicates in A297826. %t A297829 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]); %t A297829 tbl = {}; a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; %t A297829 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + n; %t A297829 b[n_] := b[n] = mex[tbl = Join[{a[n], a[n - 1], b[n - 1]}, tbl], b[n - 1]]; %t A297829 du = Differences[Table[a[n], {n, 0, 1000}]] (* A297827 *) %t A297829 w = Flatten[Position[du, 0]]; Map[a, w] (* A297829 *) %Y A297829 Cf. A297826. %K A297829 nonn,easy,new %O A297829 1,1 %A A297829 _Clark Kimberling_, Feb 04 2018 %I A297828 %S A297828 1,1,1,2,2,2,1,1,2,1,2,1,3,2,1,1,3,1,1,3,1,1,1,1,2,3,2,1,1,1,1,2,2,1, %T A297828 1,1,1,2,2,2,2,1,1,2,1,1,1,2,3,2,2,2,1,1,2,1,3,2,2,2,1,1,2,1,2,1,2,1, %U A297828 3,2,1,1,3,2,2,1,1,2,1,1,1,2,2,1,2,1 %N A297828 Difference sequence of A297997. %C A297828 Conjectures: %C A297828 (1) 2 <= a(k) <= 4 for k>=1; %C A297828 (2) if d is in {1,2,3}, then a(k) = d for infinitely many k. %H A297828 Clark Kimberling, Table of n, a(n) for n = 1..10000 %t A297828 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]); %t A297828 tbl = {}; a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; %t A297828 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + n; %t A297828 b[n_] := b[n] = mex[tbl = Join[{a[n], a[n - 1], b[n - 1]}, tbl], b[n - 1]]; %t A297828 u = Table[a[n], {n, 0, 300}](* A297826 *) %t A297828 v = Table[b[n], {n, 0, 300}](* A297997 *) %t A297828 Differences[u]; (* A297827 *) %t A297828 Differences[v]; (* A297828 *) %t A297828 (* _Peter J. C. Moses_, Jan 03 2017 *) %Y A297828 Cf. A297826, A297827. %K A297828 nonn,easy,new %O A297828 1,4 %A A297828 _Clark Kimberling_, Feb 04 2018 %I A299199 %S A299199 1,4,14,98,2,4386,18,324,60,36457092,12,5769254382,2598,78,414, %T A299199 335391687123174,510,115428139222691670,30,1926,20204166, %U A299199 24752828962220504429646,6,1032336,3124309416,149376,3816,8542182056001396008878674488976,96 %N A299199 In factorial base, any rational number has a terminating expansion; hence we can devise a self-inverse permutation of the rational numbers, say f, such that for any rational number q, the representations of q and of f(q) in factorial base are mirrored around the radix point and q and f(q) have the same sign; a(n) = f(1/n). %C A299199 See A299161 for additional comments about f. %C A299199 This sequence corresponds to the indices of ones in A299161. %H A299199 Wikipedia, Factorial number system (Fractional values) %H A299199 Index entries for sequences related to factorial base representation %H A299199 Rémy Sigrist, Colored logarithmic scatterplot of the first 5000 terms (where the color is function of A299160(n)) %F A299199 A034968(a(n)) = A276350(n) for any n > 1. %F A299199 A299160(a(n)) = n for any n > 1. %F A299199 A299161(a(n)) = 1 for any n > 1. %e A299199 The first terms, alongside the factorial base representations of a(n) and of 1/n, are: %e A299199 n a(n) fact(a(n)) fact(1/n) %e A299199 -- ---------- ----------------------- ------------ %e A299199 2 1 1 0.1 %e A299199 3 4 2 0 0.0 2 %e A299199 4 14 2 1 0 0.0 1 2 %e A299199 5 98 4 0 1 0 0.0 1 0 4 %e A299199 6 2 1 0 0.0 1 %e A299199 7 4386 6 0 2 3 0 0 0.0 0 3 2 0 6 %e A299199 8 18 3 0 0 0.0 0 3 %e A299199 9 324 2 3 2 0 0 0.0 0 2 3 2 %e A299199 10 60 2 2 0 0 0.0 0 2 2 %e A299199 11 36457092 10 0 4 1 3 5 0 2 0 0 0.0 0 2 0 5 3 1 4 0 10 %e A299199 12 12 2 0 0 0.0 0 2 %e A299199 13 5769254382 12 0 5 8 4 5 2 1 4 1 0 0 0.0 0 1 4 1 2 5 4 8 5 0 12 %e A299199 14 2598 3 3 3 1 0 0 0.0 0 1 3 3 3 %e A299199 15 78 3 1 0 0 0.0 0 1 3 %e A299199 16 414 3 2 1 0 0 0.0 0 1 2 3 %o A299199 (PARI) a(n) = my (v=0, q=1/n); for (r=2, oo, q *= r; v += floor(q) * (r-1)!; q = frac(q); if (q==0, return (v))) %Y A299199 Cf. A034968, A052126, A276350, A299160, A299161. %K A299199 nonn,base,new %O A299199 2,2 %A A299199 _Rémy Sigrist_, Feb 04 2018 %I A299161 %S A299161 0,1,1,2,1,5,1,13,5,17,3,7,1,7,1,3,5,11,1,5,7,19,11,23,1,61,7,27,41, %T A299161 101,1,11,13,43,23,53,11,71,31,91,17,37,2,19,3,4,7,29,1,31,11,41,7,17, %U A299161 7,67,9,29,47,107,1,3,4,23,13,14,17,77,37,97,19,39,1 %N A299161 In factorial base, any rational number has a terminating expansion; hence we can devise a self-inverse permutation of the rational numbers, say f, such that for any rational number q, the representations of q and of f(q) in factorial base are mirrored around the radix point and q and f(q) have the same sign; a(n) = the numerator of f(n). %C A299161 See A299160 for the corresponding denominators. %C A299161 The function f restricted to the nonnegative integers establishes a bijection from the nonnegative integers to the rational numbers q such that 0 <= q < 1, hence n -> a(n) / A299161(n) runs uniquely through all rational numbers q such that 0 <= q < 1. %C A299161 The rational numbers q = n + f(n) for some integer n are the fixed points of f. %C A299161 If two rational numbers, say p and q, have the same sign and can be added without carry in factorial base, then f(p + q) = f(p) + f(q). %H A299161 Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A299160(n)) %H A299161 Wikipedia, Factorial number system (Fractional values) %H A299161 Index entries for sequences related to factorial base representation %F A299161 a(n) < A299160(n) for any n >= 0. %F A299161 a(n!) = 1 for any n >= 0. %e A299161 The first terms, alongside f(n) and the factorial base representations of n and of f(n), are: %e A299161 n a(n) f(n) fact(n) fact(f(n)) %e A299161 -- ---- ---- ------- ---------- %e A299161 0 0 0 0 0.0 %e A299161 1 1 1/2 1 0.1 %e A299161 2 1 1/6 1 0 0.0 1 %e A299161 3 2 2/3 1 1 0.1 1 %e A299161 4 1 1/3 2 0 0.0 2 %e A299161 5 5 5/6 2 1 0.1 2 %e A299161 6 1 1/24 1 0 0 0.0 0 1 %e A299161 7 13 13/24 1 0 1 0.1 0 1 %e A299161 8 5 5/24 1 1 0 0.0 1 1 %e A299161 9 17 17/24 1 1 1 0.1 1 1 %e A299161 10 3 3/8 1 2 0 0.0 2 1 %e A299161 11 7 7/8 1 2 1 0.1 2 1 %e A299161 12 1 1/12 2 0 0 0.0 0 2 %e A299161 13 7 7/12 2 0 1 0.1 0 2 %e A299161 14 1 1/4 2 1 0 0.0 1 2 %e A299161 15 3 3/4 2 1 1 0.1 1 2 %e A299161 16 5 5/12 2 2 0 0.0 2 2 %e A299161 17 11 11/12 2 2 1 0.1 2 2 %e A299161 18 1 1/8 3 0 0 0.0 0 3 %e A299161 19 5 5/8 3 0 1 0.1 0 3 %e A299161 20 7 7/24 3 1 0 0.0 1 3 %o A299161 (PARI) a(n) = my (v=0); for (r=2, oo, if (n==0, return (numerator(v))); v += (n%r)/r!; n\=r) %Y A299161 Cf. A299160. %K A299161 nonn,base,frac,new %O A299161 0,4 %A A299161 _Rémy Sigrist_, Feb 04 2018 %I A299160 %S A299160 1,2,6,3,3,6,24,24,24,24,8,8,12,12,4,4,12,12,8,8,24,24,24,24,120,120, %T A299160 40,40,120,120,20,20,60,60,60,60,120,120,120,120,40,40,15,30,10,5,15, %U A299160 30,60,60,60,60,20,20,120,120,40,40,120,120,10,5,15,30,30,15 %N A299160 In factorial base, any rational number has a terminating expansion; hence we can devise a self-inverse permutation of the rational numbers, say f, such that for any rational number q, the representations of q and of f(q) in factorial base are mirrored around the radix point and q and f(q) have the same sign; a(n) = the denominator of f(n). %C A299160 See A299161 for the corresponding numerators and additional comments. %H A299160 Wikipedia, Factorial number system (Fractional values) %H A299160 Index entries for sequences related to factorial base representation %F A299160 a(n!) = (n+1)! for any n > 0. %e A299160 The first terms, alongside f(n) and the factorial base representations of n and of f(n), are: %e A299160 n a(n) f(n) fact(n) fact(f(n)) %e A299160 -- ---- ---- ------- ---------- %e A299160 0 1 0 0 0.0 %e A299160 1 2 1/2 1 0.1 %e A299160 2 6 1/6 1 0 0.0 1 %e A299160 3 3 2/3 1 1 0.1 1 %e A299160 4 3 1/3 2 0 0.0 2 %e A299160 5 6 5/6 2 1 0.1 2 %e A299160 6 24 1/24 1 0 0 0.0 0 1 %e A299160 7 24 13/24 1 0 1 0.1 0 1 %e A299160 8 24 5/24 1 1 0 0.0 1 1 %e A299160 9 24 17/24 1 1 1 0.1 1 1 %e A299160 10 8 3/8 1 2 0 0.0 2 1 %e A299160 11 8 7/8 1 2 1 0.1 2 1 %e A299160 12 12 1/12 2 0 0 0.0 0 2 %e A299160 13 12 7/12 2 0 1 0.1 0 2 %e A299160 14 4 1/4 2 1 0 0.0 1 2 %e A299160 15 4 3/4 2 1 1 0.1 1 2 %e A299160 16 12 5/12 2 2 0 0.0 2 2 %e A299160 17 12 11/12 2 2 1 0.1 2 2 %e A299160 18 8 1/8 3 0 0 0.0 0 3 %e A299160 19 8 5/8 3 0 1 0.1 0 3 %e A299160 20 24 7/24 3 1 0 0.0 1 3 %o A299160 (PARI) a(n) = my (v=0); for (r=2, oo, if (n==0, return (denominator(v))); v += (n%r)/r!; n\=r) %Y A299160 Cf. A299161. %K A299160 nonn,base,frac,new %O A299160 0,2 %A A299160 _Rémy Sigrist_, Feb 04 2018 %I A298484 %S A298484 1,2,2,3,6,6,4,12,42,42,5,20,156,1806,1806,6,30,420,24492,3263442, %T A298484 3263442,7,42,930,176820,599882556,10650056950806,10650056950806,8,56, %U A298484 1806,865830,31265489220,359859081592975692,113423713055421844361000442,113423713055421844361000442 %N A298484 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is defined by the following: A(1,k) = k and A(n,k) = A(n-1,k)*(A(n-1,k)+1) for n > 1. %e A298484 Square array begins: %e A298484 1, 2, 3, 4, 5, 6, ... %e A298484 2, 6, 12, 20, 30, 42, ... %e A298484 6, 42, 156, 420, 930, 1806, ... %e A298484 42, 1806, 24492, 176820, 865830, 3263442, ... %e A298484 1806, 3263442, 599882556, 31265489220, 74966245730, 10650056950806, ... %t A298484 A[1, k_?Positive] := k; A[n_?Positive, k_?Positive] := A[n, k] = A[n - 1, k]*(A[n - 1, k] + 1); Table[A[n - k + 1, k], {n, 1, 9}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Feb 05 2018 *) %Y A298484 Columns k=1..3 give A007018(n-1), A007018, A004168(n-1). %Y A298484 Rows n=1..3 give A000027, A002378, A169938. %K A298484 nonn,tabl,new %O A298484 1,2 %A A298484 _Seiichi Manyama_, Jan 20 2018 %I A299321 %S A299321 1,1,1,1,1,1,1,1,1,1,1,2,15,2,1,1,5,10,10,5,1,1,9,70,12,70,9,1,1,22, %T A299321 146,130,130,146,22,1,1,45,434,284,1306,284,434,45,1,1,101,1206,1557, %U A299321 6051,6051,1557,1206,101,1,1,218,3228,5838,38035,44381,38035,5838,3228,218,1 %N A299321 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299321 Table starts %C A299321 .1...1....1.....1.......1........1..........1............1.............1 %C A299321 .1...1....1.....2.......5........9.........22...........45...........101 %C A299321 .1...1...15....10......70......146........434.........1206..........3228 %C A299321 .1...2...10....12.....130......284.......1557.........5838.........24821 %C A299321 .1...5...70...130....1306.....6051......38035.......234329.......1457212 %C A299321 .1...9..146...284....6051....44381.....473924......4898412......51303757 %C A299321 .1..22..434..1557...38035...473924....7885194....132676384....2249698337 %C A299321 .1..45.1206..5838..234329..4898412..132676384...3745566028..105240244212 %C A299321 .1.101.3228.24821.1457212.51303757.2249698337.105240244212.4909629881306 %H A299321 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299321 Empirical for column k: %F A299321 k=1: a(n) = a(n-1) %F A299321 k=2: a(n) = a(n-1) +3*a(n-2) -2*a(n-4) for n>5 %F A299321 k=3: [order 10] for n>12 %F A299321 k=4: [order 29] for n>30 %e A299321 Some solutions for n=5 k=4 %e A299321 ..0..0..0..1. .0..1..1..0. .0..0..0..0. .0..0..0..0. .0..1..1..0 %e A299321 ..0..0..0..0. .1..1..1..1. .0..0..0..0. .0..0..0..0. .1..1..1..1 %e A299321 ..1..1..0..0. .0..1..1..0. .1..0..1..0. .0..1..0..1. .0..1..1..1 %e A299321 ..1..1..1..1. .1..1..1..1. .0..0..0..0. .0..0..0..0. .1..1..0..0 %e A299321 ..0..1..1..1. .0..1..1..1. .0..0..0..0. .1..0..0..1. .1..1..0..0 %Y A299321 Column 2 is A052962(n-2). %K A299321 nonn,tabl,new %O A299321 1,12 %A A299321 _R. H. Hardin_, Feb 06 2018 %I A299320 %S A299320 1,22,434,1557,38035,473924,7885194,132676384,2249698337,39854864168, %T A299320 700615705103,12435929820498,221099245989227,3935763610862580, %U A299320 70136675735762945,1250168095233297669,22291499935243883463 %N A299320 Number of nX7 0..1 arrays with every element equal to 0, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299320 Column 7 of A299321. %H A299320 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299320 Some solutions for n=5 %e A299320 ..0..1..1..1..1..1..1. .0..1..1..1..0..1..1. .0..1..1..1..0..0..1 %e A299320 ..1..1..0..0..1..1..1. .1..1..0..1..1..1..1. .1..1..1..1..0..0..0 %e A299320 ..0..1..0..0..1..0..0. .1..1..1..1..0..0..0. .1..1..1..0..0..0..1 %e A299320 ..1..1..1..1..0..0..0. .0..0..1..0..0..0..0. .0..0..0..0..1..0..0 %e A299320 ..0..1..1..1..0..0..0. .0..0..0..0..1..0..1. .0..0..1..0..0..0..0 %Y A299320 Cf. A299321. %K A299320 nonn,new %O A299320 1,2 %A A299320 _R. H. Hardin_, Feb 06 2018 %I A299319 %S A299319 1,9,146,284,6051,44381,473924,4898412,51303757,560144415,6042198393, %T A299319 65973666490,720410978977,7878701955428,86252362060965, %U A299319 944390263513343,10344434170986780,113317722430202816,1241460522814886751 %N A299319 Number of nX6 0..1 arrays with every element equal to 0, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299319 Column 6 of A299321. %H A299319 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299319 Some solutions for n=5 %e A299319 ..0..0..1..1..1..1. .0..1..1..0..0..1. .0..0..0..1..0..0. .0..0..0..0..0..1 %e A299319 ..0..0..1..1..1..1. .1..1..1..0..0..0. .0..0..0..0..0..0. .0..0..0..1..0..0 %e A299319 ..0..0..1..1..0..1. .1..0..1..0..0..1. .0..0..1..0..1..1. .1..0..0..0..0..0 %e A299319 ..0..0..0..1..1..1. .1..1..1..0..0..0. .1..1..1..1..1..1. .0..0..0..1..0..0 %e A299319 ..0..0..0..1..1..1. .1..1..1..0..0..0. .1..1..1..0..1..1. .0..0..0..0..0..0 %Y A299319 Cf. A299321. %K A299319 nonn,new %O A299319 1,2 %A A299319 _R. H. Hardin_, Feb 06 2018 %I A299318 %S A299318 1,5,70,130,1306,6051,38035,234329,1457212,9460174,61098363,400690315, %T A299318 2635532413,17400923491,115170103841,763257987419,5063822195573, %U A299318 33615913766700,223256247032907,1483145326036963,9854734637542620 %N A299318 Number of nX5 0..1 arrays with every element equal to 0, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299318 Column 5 of A299321. %H A299318 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299318 Some solutions for n=5 %e A299318 ..0..0..1..0..0. .0..1..0..1..1. .0..1..1..1..0. .0..1..0..1..1 %e A299318 ..0..0..0..0..0. .1..1..1..1..1. .1..1..1..1..1. .1..1..1..1..1 %e A299318 ..0..0..1..1..1. .0..1..1..1..0. .1..1..0..1..0. .0..1..1..1..0 %e A299318 ..1..1..1..1..1. .1..1..0..1..1. .1..1..1..1..1. .1..1..0..1..1 %e A299318 ..1..1..0..1..1. .1..1..1..1..1. .1..1..0..1..0. .0..1..1..1..0 %Y A299318 Cf. A299321. %K A299318 nonn,new %O A299318 1,2 %A A299318 _R. H. Hardin_, Feb 06 2018 %I A299317 %S A299317 1,2,10,12,130,284,1557,5838,24821,106561,449606,1956599,8438660, %T A299317 36764273,160295868,700439879,3066084680,13431404246,58893058333, %U A299317 258348784769,1133779588694,4977030114598,21852290559017,95959325703386 %N A299317 Number of nX4 0..1 arrays with every element equal to 0, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299317 Column 4 of A299321. %H A299317 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299317 Empirical: a(n) = 3*a(n-1) +15*a(n-2) -12*a(n-3) -117*a(n-4) -82*a(n-5) +220*a(n-6) +455*a(n-7) +187*a(n-8) -278*a(n-9) -592*a(n-10) -695*a(n-11) -353*a(n-12) +604*a(n-13) +404*a(n-14) -92*a(n-15) +874*a(n-16) +845*a(n-17) -523*a(n-18) -728*a(n-19) -234*a(n-20) -26*a(n-21) +90*a(n-22) +42*a(n-23) -39*a(n-24) +15*a(n-25) +40*a(n-26) +4*a(n-27) -8*a(n-28) -2*a(n-29) for n>30 %e A299317 Some solutions for n=5 %e A299317 ..0..0..0..1. .0..1..1..0. .0..0..1..1. .0..0..0..0. .0..0..0..1 %e A299317 ..0..0..0..0. .1..1..1..1. .0..0..1..1. .0..0..0..0. .0..0..0..0 %e A299317 ..1..0..0..1. .1..0..1..1. .1..1..1..0. .1..0..0..1. .1..0..0..0 %e A299317 ..0..0..0..0. .1..1..1..1. .1..1..1..1. .0..0..0..0. .0..0..1..1 %e A299317 ..1..0..0..1. .1..1..1..1. .0..1..1..1. .0..0..0..0. .0..0..1..1 %Y A299317 Cf. A299321. %K A299317 nonn,new %O A299317 1,2 %A A299317 _R. H. Hardin_, Feb 06 2018 %I A299316 %S A299316 1,1,15,10,70,146,434,1206,3228,9186,24772,69398,189942,526396, %T A299316 1450524,4005122,11055658,30506770,84220672,232416524,641554400, %U A299316 1770641470,4887237832,13489036144,37230869642,102760304362,283626021494 %N A299316 Number of nX3 0..1 arrays with every element equal to 0, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299316 Column 3 of A299321. %H A299316 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299316 Empirical: a(n) = 2*a(n-1) +5*a(n-2) -5*a(n-3) -10*a(n-4) +4*a(n-5) +a(n-6) +3*a(n-7) -3*a(n-8) +4*a(n-9) -2*a(n-10) for n>12 %e A299316 Some solutions for n=5 %e A299316 ..0..0..1. .0..0..0. .0..1..0. .0..0..0. .0..0..1. .0..1..1. .0..1..0 %e A299316 ..0..0..0. .0..0..0. .1..1..1. .0..0..0. .0..0..0. .1..1..1. .1..1..1 %e A299316 ..0..0..0. .1..0..1. .0..1..0. .0..0..1. .0..1..0. .0..1..1. .1..0..1 %e A299316 ..1..1..1. .0..0..0. .1..1..1. .0..0..0. .0..0..0. .1..1..1. .1..1..1 %e A299316 ..1..1..1. .0..0..0. .0..1..1. .1..0..1. .1..0..0. .0..1..0. .1..1..1 %Y A299316 Cf. A299321. %K A299316 nonn,new %O A299316 1,3 %A A299316 _R. H. Hardin_, Feb 06 2018 %I A299315 %S A299315 1,1,15,12,1306,44381,7885194,3745566028,4909629881306 %N A299315 Number of nXn 0..1 arrays with every element equal to 0, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299315 Diagonal of A299321. %e A299315 Some solutions for n=5 %e A299315 ..0..0..0..0..1. .0..0..0..1..1. .0..1..1..1..1. .0..0..1..1..1 %e A299315 ..0..0..0..0..0. .0..0..0..1..1. .1..1..1..1..1. .0..0..1..1..1 %e A299315 ..1..0..1..0..0. .0..0..1..1..0. .0..1..0..1..0. .0..0..0..1..1 %e A299315 ..0..0..0..0..0. .0..0..0..1..1. .1..1..1..1..1. .0..0..0..0..0 %e A299315 ..0..0..0..0..1. .1..0..0..1..1. .1..1..1..1..1. .1..0..1..0..0 %Y A299315 Cf. A299321. %K A299315 nonn,new %O A299315 1,3 %A A299315 _R. H. Hardin_, Feb 06 2018 %I A299314 %S A299314 0,1,1,1,3,1,2,7,7,2,3,13,15,13,3,5,23,29,29,23,5,8,49,63,112,63,49,8, %T A299314 13,99,199,504,504,199,99,13,21,189,593,2528,3463,2528,593,189,21,34, %U A299314 383,1657,11252,24519,24519,11252,1657,383,34,55,777,4689,50720,167810 %N A299314 T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299314 Table starts %C A299314 ..0...1....1......2.......3.........5...........8...........13.............21 %C A299314 ..1...3....7.....13......23........49..........99..........189............383 %C A299314 ..1...7...15.....29......63.......199.........593.........1657...........4689 %C A299314 ..2..13...29....112.....504......2528.......11252........50720.........241309 %C A299314 ..3..23...63....504....3463.....24519......167810......1165033........8305148 %C A299314 ..5..49..199...2528...24519....270346.....3203795.....36378590......417642077 %C A299314 ..8..99..593..11252..167810...3203795....60731972...1089584523....20249517439 %C A299314 .13.189.1657..50720.1165033..36378590..1089584523..30975700202...924197633148 %C A299314 .21.383.4689.241309.8305148.417642077.20249517439.924197633148.44405195921475 %H A299314 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299314 Empirical for column k: %F A299314 k=1: a(n) = a(n-1) +a(n-2) %F A299314 k=2: a(n) = 2*a(n-1) -a(n-2) +4*a(n-3) -4*a(n-4) for n>5 %F A299314 k=3: [order 16] for n>17 %F A299314 k=4: [order 62] for n>66 %e A299314 Some solutions for n=5 k=4 %e A299314 ..0..1..0..0. .0..0..1..0. .0..1..1..0. .0..0..0..0. .0..1..1..0 %e A299314 ..0..1..1..1. .1..1..1..0. .0..0..0..1. .1..0..0..1. .0..0..1..0 %e A299314 ..0..1..1..0. .0..0..1..1. .0..0..0..1. .0..1..0..1. .0..0..1..1 %e A299314 ..0..1..1..0. .1..0..1..1. .1..0..0..1. .1..0..0..1. .0..0..1..1 %e A299314 ..0..1..0..1. .0..1..0..1. .0..1..0..1. .1..0..1..0. .0..1..0..1 %Y A299314 Column 1 is A000045(n-1). %Y A299314 Column 2 is A297953. %K A299314 nonn,tabl,new %O A299314 1,5 %A A299314 _R. H. Hardin_, Feb 06 2018 %I A299313 %S A299313 8,99,593,11252,167810,3203795,60731972,1089584523,20249517439, %T A299313 376576842172,6933812025046,128217475610989,2373760598011141, %U A299313 43877591299939779,811240029844441755,15004022659152917520 %N A299313 Number of nX7 0..1 arrays with every element equal to 1, 2, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299313 Column 7 of A299314. %H A299313 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299313 Some solutions for n=5 %e A299313 ..0..0..1..1..0..1..1. .0..1..0..1..1..1..1. .0..0..1..1..1..1..0 %e A299313 ..0..1..1..1..1..0..0. .1..0..1..0..0..0..0. .1..1..0..0..0..0..1 %e A299313 ..0..0..0..1..1..1..1. .0..1..0..1..0..0..1. .0..0..0..0..0..1..0 %e A299313 ..0..0..0..0..0..0..1. .0..1..1..1..1..0..1. .0..1..1..0..0..1..0 %e A299313 ..0..1..1..0..0..1..1. .1..1..1..1..0..1..0. .1..1..1..1..0..1..0 %Y A299313 Cf. A299314. %K A299313 nonn,new %O A299313 1,1 %A A299313 _R. H. Hardin_, Feb 06 2018 %I A299312 %S A299312 5,49,199,2528,24519,270346,3203795,36378590,417642077,4822467446, %T A299312 55430028860,637788505212,7344874274460,84535258092563, %U A299312 972944766688659,11199796394782944,128917185100141603,1483879490840389393 %N A299312 Number of nX6 0..1 arrays with every element equal to 1, 2, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299312 Column 6 of A299314. %H A299312 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299312 Some solutions for n=5 %e A299312 ..0..0..1..1..1..0. .0..1..0..0..1..0. .0..1..1..1..1..0. .0..0..0..1..0..1 %e A299312 ..1..0..1..0..1..0. .1..0..1..1..0..1. .0..0..1..1..0..0. .1..1..1..0..1..0 %e A299312 ..0..1..0..1..1..1. .1..0..0..0..0..1. .0..0..0..0..0..0. .0..1..1..1..1..1 %e A299312 ..1..1..1..1..0..0. .1..0..0..0..0..0. .1..0..1..0..0..0. .1..0..0..1..0..0 %e A299312 ..0..0..1..1..1..0. .0..1..0..0..1..1. .0..1..0..1..1..0. .0..1..0..1..1..0 %Y A299312 Cf. A299314. %K A299312 nonn,new %O A299312 1,1 %A A299312 _R. H. Hardin_, Feb 06 2018 %I A299311 %S A299311 3,23,63,504,3463,24519,167810,1165033,8305148,58382562,409804187, %T A299311 2900191611,20484580351,144448486221,1020055338337,7204564257340, %U A299311 50860269292839,359093840742852,2535713227786876,17904201238729402 %N A299311 Number of nX5 0..1 arrays with every element equal to 1, 2, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299311 Column 5 of A299314. %H A299311 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299311 Some solutions for n=5 %e A299311 ..0..1..1..0..1. .0..1..1..1..1. .0..0..1..1..0. .0..0..0..0..1 %e A299311 ..0..0..0..0..1. .1..0..0..0..0. .1..1..0..0..1. .1..1..1..1..1 %e A299311 ..0..0..0..0..0. .1..0..0..0..1. .0..0..0..0..1. .0..1..1..1..1 %e A299311 ..1..0..1..0..0. .1..0..1..0..1. .1..0..0..0..1. .0..0..0..0..1 %e A299311 ..1..0..1..1..0. .0..1..0..1..0. .1..0..1..1..0. .1..1..0..0..0 %Y A299311 Cf. A299314. %K A299311 nonn,new %O A299311 1,1 %A A299311 _R. H. Hardin_, Feb 06 2018 %I A299310 %S A299310 2,13,29,112,504,2528,11252,50720,241309,1120649,5164117,24001749, %T A299310 111611790,517587900,2401369229,11150079663,51754367711,240183038282, %U A299310 1114814806501,5174545144732,24016953668166,111472218446769 %N A299310 Number of nX4 0..1 arrays with every element equal to 1, 2, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299310 Column 4 of A299314. %H A299310 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299310 Empirical: a(n) = 6*a(n-1) -9*a(n-2) +24*a(n-3) -47*a(n-4) -33*a(n-5) +67*a(n-6) -44*a(n-7) -1266*a(n-8) +1618*a(n-9) -650*a(n-10) +8875*a(n-11) -21535*a(n-12) +45298*a(n-13) -28362*a(n-14) +8063*a(n-15) -33255*a(n-16) +137036*a(n-17) -475039*a(n-18) +685036*a(n-19) -1381737*a(n-20) +2122260*a(n-21) -2994250*a(n-22) +3090285*a(n-23) -1926929*a(n-24) +3223180*a(n-25) -4173823*a(n-26) +4989813*a(n-27) +3861224*a(n-28) -12392223*a(n-29) +10955676*a(n-30) -4368770*a(n-31) +6543615*a(n-32) -28383782*a(n-33) +21197884*a(n-34) -9458914*a(n-35) -3445614*a(n-36) -34122171*a(n-37) +61493313*a(n-38) -7568366*a(n-39) -21654985*a(n-40) +47622275*a(n-41) -5153772*a(n-42) +1833710*a(n-43) -29408089*a(n-44) +26364129*a(n-45) -23403481*a(n-46) -25753860*a(n-47) -1475594*a(n-48) +28400558*a(n-49) -453526*a(n-50) -4211936*a(n-51) -3649669*a(n-52) -41937*a(n-53) +2293068*a(n-54) +1341106*a(n-55) +369512*a(n-56) -276982*a(n-57) -141152*a(n-58) -41792*a(n-59) -9252*a(n-60) +336*a(n-61) +148*a(n-62) for n>66 %e A299310 Some solutions for n=5 %e A299310 ..0..1..0..0. .0..1..0..0. .0..1..1..0. .0..1..1..1. .0..1..0..1 %e A299310 ..0..1..1..0. .0..0..1..1. .1..0..0..1. .0..0..0..0. .0..0..1..1 %e A299310 ..1..1..1..1. .0..0..0..0. .0..0..0..0. .0..0..0..1. .1..1..0..0 %e A299310 ..0..0..1..1. .0..0..0..1. .1..1..0..0. .1..0..0..1. .0..1..1..1 %e A299310 ..1..1..0..1. .0..1..1..0. .0..0..1..0. .0..1..1..0. .1..0..0..1 %Y A299310 Cf. A299314. %K A299310 nonn,new %O A299310 1,1 %A A299310 _R. H. Hardin_, Feb 06 2018 %I A299309 %S A299309 1,7,15,29,63,199,593,1657,4689,13395,38357,109855,313667,896417, %T A299309 2563517,7330205,20956899,59914557,171302635,489779503,1400327457, %U A299309 4003663145,11446878297,32727841823,93572301793,267532776375,764903730599 %N A299309 Number of nX3 0..1 arrays with every element equal to 1, 2, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299309 Column 3 of A299314. %H A299309 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299309 Empirical: a(n) = 4*a(n-1) -3*a(n-2) +a(n-3) -3*a(n-4) -6*a(n-5) -8*a(n-6) +18*a(n-7) +27*a(n-8) -18*a(n-9) +15*a(n-10) -55*a(n-11) +21*a(n-12) -6*a(n-13) +14*a(n-14) +2*a(n-15) -2*a(n-16) for n>17 %e A299309 Some solutions for n=6 %e A299309 ..0..1..1. .0..1..0. .0..1..0. .0..1..0. .0..1..0. .0..1..1. .0..1..1 %e A299309 ..0..0..1. .1..0..0. .1..0..1. .0..1..0. .0..0..1. .0..0..1. .0..0..1 %e A299309 ..0..0..0. .0..0..0. .0..0..0. .1..1..1. .0..0..0. .0..0..0. .0..0..0 %e A299309 ..0..0..0. .0..0..0. .0..0..0. .1..1..1. .0..0..0. .0..0..0. .0..0..0 %e A299309 ..1..0..1. .1..0..1. .0..0..1. .1..1..0. .1..0..0. .1..0..0. .0..0..1 %e A299309 ..1..0..1. .0..1..0. .0..1..0. .1..0..0. .0..1..0. .0..1..0. .0..1..1 %Y A299309 Cf. A299314. %K A299309 nonn,new %O A299309 1,2 %A A299309 _R. H. Hardin_, Feb 06 2018 %I A299308 %S A299308 0,3,15,112,3463,270346,60731972,30975700202,44405195921475 %N A299308 Number of nXn 0..1 arrays with every element equal to 1, 2, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299308 Diagonal of A299314. %e A299308 Some solutions for n=5 %e A299308 ..0..1..1..1..0. .0..1..1..1..0. .0..1..1..1..0. .0..1..0..1..0 %e A299308 ..0..1..0..1..0. .0..1..0..1..0. .1..0..0..0..1. .0..1..1..0..1 %e A299308 ..1..0..1..0..1. .1..0..1..0..1. .1..0..0..0..1. .1..0..1..0..0 %e A299308 ..1..0..0..0..0. .0..1..0..1..0. .0..0..1..0..1. .0..1..1..0..0 %e A299308 ..0..0..0..1..1. .1..0..1..0..1. .1..1..0..1..0. .0..1..0..1..0 %Y A299308 Cf. A299314. %K A299308 nonn,new %O A299308 1,2 %A A299308 _R. H. Hardin_, Feb 06 2018 %I A299307 %S A299307 0,1,1,1,4,1,2,18,18,2,3,52,56,52,3,5,174,223,223,174,5,8,604,849, %T A299307 1024,849,604,8,13,2048,3387,5360,5360,3387,2048,13,21,6948,13075, %U A299307 28374,55390,28374,13075,6948,21,34,23652,51006,144040,482418,482418,144040,51006 %N A299307 T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299307 Table starts %C A299307 ..0.....1......1.......2.........3...........5............8.............13 %C A299307 ..1.....4.....18......52.......174.........604.........2048...........6948 %C A299307 ..1....18.....56.....223.......849........3387........13075..........51006 %C A299307 ..2....52....223....1024......5360.......28374.......144040.........743640 %C A299307 ..3...174....849....5360.....55390......482418......3860858.......34734150 %C A299307 ..5...604...3387...28374....482418.....6525374.....76611604.....1041660250 %C A299307 ..8..2048..13075..144040...3860858....76611604...1227812226....23819382545 %C A299307 .13..6948..51006..743640..34734150..1041660250..23819382545...723886147451 %C A299307 .21.23652.199243.3857242.308990628.14216373088.471369052777.22415064379413 %H A299307 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299307 Empirical for column k: %F A299307 k=1: a(n) = a(n-1) +a(n-2) %F A299307 k=2: a(n) = 4*a(n-1) -2*a(n-2) +2*a(n-3) -6*a(n-4) -4*a(n-5) for n>6 %F A299307 k=3: [order 20] for n>21 %F A299307 k=4: [order 68] for n>70 %e A299307 Some solutions for n=5 k=4 %e A299307 ..0..0..1..1. .0..1..0..0. .0..1..0..1. .0..1..1..1. .0..0..1..0 %e A299307 ..1..0..1..0. .0..1..0..1. .0..1..1..0. .1..0..0..1. .1..1..0..1 %e A299307 ..1..1..0..0. .0..1..0..1. .0..1..0..1. .0..1..1..0. .1..1..1..1 %e A299307 ..0..1..1..1. .0..1..0..1. .0..0..0..1. .1..0..0..1. .1..1..1..0 %e A299307 ..0..1..1..1. .1..1..0..0. .0..1..1..1. .1..1..0..1. .0..0..1..0 %Y A299307 Column 1 is A000045(n-1). %Y A299307 Column 2 is A297945. %Y A299307 Column 3 is A298384. %K A299307 nonn,tabl,new %O A299307 1,5 %A A299307 _R. H. Hardin_, Feb 06 2018 %I A299306 %S A299306 8,2048,13075,144040,3860858,76611604,1227812226,23819382545, %T A299306 471369052777,8749447076371,164571560365968,3152188315142786, %U A299306 59832211663862100,1132092731063667371,21504285978321722965 %N A299306 Number of nX7 0..1 arrays with every element equal to 1, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299306 Column 7 of A299307. %H A299306 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299306 Some solutions for n=5 %e A299306 ..0..1..0..1..0..0..1. .0..1..0..1..1..1..1. .0..1..1..1..0..1..0 %e A299306 ..0..1..0..1..1..0..1. .0..1..0..0..1..0..0. .0..1..0..0..0..0..1 %e A299306 ..0..1..1..1..1..1..0. .0..1..1..1..1..1..1. .0..1..0..0..0..0..0 %e A299306 ..0..1..0..1..1..0..1. .0..1..0..1..1..0..1. .1..0..1..0..0..1..1 %e A299306 ..0..0..1..0..0..1..0. .0..0..1..1..0..1..0. .0..0..0..1..1..0..1 %Y A299306 Cf. A299307. %K A299306 nonn,new %O A299306 1,1 %A A299306 _R. H. Hardin_, Feb 06 2018 %I A299305 %S A299305 5,604,3387,28374,482418,6525374,76611604,1041660250,14216373088, %T A299305 185641725091,2461636096556,32972981437712,438191632134687, %U A299305 5821100136764371,77544942985407707,1032213930246913442 %N A299305 Number of nX6 0..1 arrays with every element equal to 1, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299305 Column 6 of A299307. %H A299305 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299305 Some solutions for n=5 %e A299305 ..0..1..1..0..0..1. .0..0..1..1..1..1. .0..0..0..1..0..0. .0..1..1..0..0..1 %e A299305 ..0..0..0..1..1..1. .1..0..0..0..0..1. .1..0..1..1..0..1. .0..0..1..1..1..0 %e A299305 ..1..0..0..0..0..0. .0..1..0..0..0..1. .1..1..1..1..1..1. .0..1..1..0..1..0 %e A299305 ..1..0..1..1..1..0. .0..0..0..0..1..0. .0..0..1..1..1..0. .1..0..0..0..1..0 %e A299305 ..0..1..1..0..0..1. .0..1..1..0..1..1. .0..0..1..0..0..1. .0..1..1..1..0..0 %Y A299305 Cf. A299307. %K A299305 nonn,new %O A299305 1,1 %A A299305 _R. H. Hardin_, Feb 06 2018 %I A299304 %S A299304 3,174,849,5360,55390,482418,3860858,34734150,308990628,2666332423, %T A299304 23444806458,206971861916,1813834330882,15927556058496, %U A299304 140120710747692,1231062726144171,10815430485747421,95064758912339098,835449813794549847 %N A299304 Number of nX5 0..1 arrays with every element equal to 1, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299304 Column 5 of A299307. %H A299304 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299304 Some solutions for n=5 %e A299304 ..0..1..0..1..0. .0..0..1..0..0. .0..1..0..1..1. .0..1..0..0..0 %e A299304 ..1..0..0..0..1. .1..0..1..1..1. .0..1..0..0..0. .0..1..1..0..1 %e A299304 ..0..1..0..0..0. .1..1..1..1..1. .0..0..0..0..1. .1..0..0..0..1 %e A299304 ..1..1..0..0..1. .0..1..0..1..1. .1..0..1..1..0. .1..1..1..1..0 %e A299304 ..0..0..1..0..1. .0..1..1..0..0. .0..1..0..0..0. .0..0..0..1..0 %Y A299304 Cf. A299307. %K A299304 nonn,new %O A299304 1,1 %A A299304 _R. H. Hardin_, Feb 06 2018 %I A299303 %S A299303 2,52,223,1024,5360,28374,144040,743640,3857242,19925599,102977350, %T A299303 532709362,2754912665,14246433006,73680910336,381067786735, %U A299303 1970816457786,10192855468940,52716573704127,272645676769182,1410103205704342 %N A299303 Number of nX4 0..1 arrays with every element equal to 1, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299303 Column 4 of A299307. %H A299303 R. H. Hardin, Table of n, a(n) for n = 1..210 %H A299303 R. H. Hardin, Empirical recurrence of order 68 %F A299303 Empirical recurrence of order 68 (see link above) %e A299303 Some solutions for n=5 %e A299303 ..0..0..1..1. .0..0..0..0. .0..0..0..1. .0..1..1..1. .0..0..1..0 %e A299303 ..0..0..1..1. .0..1..1..1. .1..0..1..1. .0..1..1..1. .1..1..0..1 %e A299303 ..0..0..0..0. .0..1..1..1. .0..1..0..0. .1..0..1..1. .1..1..1..1 %e A299303 ..0..0..0..1. .1..0..1..1. .1..0..0..0. .1..0..1..1. .1..1..1..0 %e A299303 ..1..1..1..0. .1..0..1..1. .1..0..0..0. .1..0..1..1. .0..0..0..1 %Y A299303 Cf. A299307. %K A299303 nonn,new %O A299303 1,1 %A A299303 _R. H. Hardin_, Feb 06 2018 %I A299302 %S A299302 0,4,56,1024,55390,6525374,1227812226,723886147451,1084184652392138 %N A299302 Number of nXn 0..1 arrays with every element equal to 1, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299302 Diagonal of A299307. %e A299302 Some solutions for n=5 %e A299302 ..0..0..1..1..0. .0..0..1..0..0. .0..0..1..1..0. .0..0..1..0..1 %e A299302 ..0..1..0..0..0. .1..0..0..1..1. .0..0..1..0..1. .1..1..1..1..0 %e A299302 ..1..1..0..1..0. .1..0..0..0..0. .0..0..1..0..1. .1..1..1..1..0 %e A299302 ..0..1..0..1..0. .0..0..0..1..0. .0..0..1..0..0. .1..1..0..1..1 %e A299302 ..0..0..1..0..1. .1..1..0..0..1. .0..0..1..1..1. .0..0..1..0..1 %Y A299302 Cf. A299307. %K A299302 nonn,new %O A299302 1,2 %A A299302 _R. H. Hardin_, Feb 06 2018 %I A299253 %S A299253 1,3,4,6,8,12,16,24,32,48,64,96,126,183,242,357,472,696,920,1356,1792, %T A299253 2640,3486,5136,6788,10002,13216,19473,25730,37911,50092,73806,97518, %U A299253 143688,189860,279744,369628,544620,719612,1060296,1400980,2064243,2727504,4018785,5310068,7824000,10337932,15232200,20126468 %N A299253 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^12 = 1 >. %H A299253 Colin Barker, Table of n, a(n) for n = 0..1000 %H A299253 Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,2,0,3,0,5,0,3,0,2,0,1,0,0,0,-1). %F A299253 G.f.: (-2*x^22 + 3*x^20 + 3*x^19 + 6*x^18 + 6*x^17 + 9*x^16 + 9*x^15 + 12*x^14 + 12*x^13 + 15*x^12 + 15*x^11 + 15*x^10 + 15*x^9 + 13*x^8 + 12*x^7 + 10*x^6 + 9*x^5 + 7*x^4 + 6*x^3 + 4*x^2 + 3*x + 1)/(x^20 - x^16 - 2*x^14 - 3*x^12 - 5*x^10 - 3*x^8 - 2*x^6 - x^4 + 1). %F A299253 a(n) = a(n-4) + 2*a(n-6) + 3*a(n-8) + 5*a(n-10) + 3*a(n-12) + 2*a(n-14) + a(n-16) - a(n-20) for n>20. - _Colin Barker_, Feb 06 2018 %o A299253 (MAGMA) See Magma program in A298805. %o A299253 (PARI) Vec((1 + 3*x + 4*x^2 + 6*x^3 + 7*x^4 + 9*x^5 + 10*x^6 + 12*x^7 + 13*x^8 + 15*x^9 + 15*x^10 + 15*x^11 + 15*x^12 + 12*x^13 + 12*x^14 + 9*x^15 + 9*x^16 + 6*x^17 + 6*x^18 + 3*x^19 + 3*x^20 - 2*x^22) / ((1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)*(1 - x^2 - x^4 - x^6 - x^8 - x^10 + x^12)) + O(x^60)) \\ _Colin Barker_, Feb 06 2018 %Y A299253 Cf. A008579, A298802, A298805. %K A299253 nonn,easy,new %O A299253 0,2 %A A299253 _John Cannon_ and _N. J. A. Sloane_, Feb 06 2018 %I A299252 %S A299252 1,3,4,6,8,12,16,24,32,48,64,94,120,178,232,344,448,664,864,1280,1662, %T A299252 2459,3202,4741,6168,9132,11880,17588,22880,33870,44068,65246,84880, %U A299252 125664,163484,242036,314880,466176,606478,897892,1168124,1729394,2249880,3330929,4333418,6415591,8346452,12356856,16075828,23800132 %N A299252 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^11 = 1 >. %H A299252 Colin Barker, Table of n, a(n) for n = 0..1000 %H A299252 Index entries for linear recurrences with constant coefficients, signature (-1,0,0,1,1,2,2,3,4,3,2,3,2,3,2,3,2,3,4,3,2,2,1,1,0,0,-1,-1). %F A299252 G.f.: (-2*x^30 - 2*x^29 + 3*x^28 + 6*x^27 + 9*x^26 + 12*x^25 + 15*x^24 + 18*x^23 + 21*x^22 + 24*x^21 + 27*x^20 + 31*x^19 + 33*x^18 + 33*x^17 + 33*x^16 + 33*x^15 + 33*x^14 + 33*x^13 + 33*x^12 + 33*x^11 + 29*x^10 + 27*x^9 + 25*x^8 + 22*x^7 + 19*x^6 + 16*x^5 + 13*x^4 + 10*x^3 + 7*x^2 + 4*x + 1)/(x^28 + x^27 - x^24 - x^23 - 2*x^22 - 2*x^21 - 3*x^20 - 4*x^19 - 3*x^18 - 2*x^17 - 3*x^16 - 2*x^15 - 3*x^14 - 2*x^13 - 3*x^12 - 2*x^11 - 3*x^10 - 4*x^9 - 3*x^8 - 2*x^7 - 2*x^6 - x^5 - x^4 + x + 1). %F A299252 a(n) = -a(n-1) + a(n-4) + a(n-5) + 2*a(n-6) + 2*a(n-7) + 3*a(n-8) + 4*a(n-9) + 3*a(n-10) + 2*a(n-11) + 3*a(n-12) + 2*a(n-13) + 3*a(n-14) + 2*a(n-15) + 3*a(n-16) + 2*a(n-17) + 3*a(n-18) + 4*a(n-19) + 3*a(n-20) + 2*a(n-21) + 2*a(n-22) + a(n-23) + a(n-24) - a(n-27) - a(n-28) for n>30. - _Colin Barker_, Feb 06 2018 %o A299252 (MAGMA) See Magma program in A298805. %o A299252 (PARI) Vec((1 + 4*x + 7*x^2 + 10*x^3 + 13*x^4 + 16*x^5 + 19*x^6 + 22*x^7 + 25*x^8 + 27*x^9 + 29*x^10 + 33*x^11 + 33*x^12 + 33*x^13 + 33*x^14 + 33*x^15 + 33*x^16 + 33*x^17 + 33*x^18 + 31*x^19 + 27*x^20 + 24*x^21 + 21*x^22 + 18*x^23 + 15*x^24 + 12*x^25 + 9*x^26 + 6*x^27 + 3*x^28 - 2*x^29 - 2*x^30) / ((1 + x + x^2)*(1 + x^3 + x^6)*(1 - x^2 - x^4 - x^6 - x^8 + x^10 - x^12 - x^14 - x^16 - x^18 + x^20)) + O(x^60)) \\ _Colin Barker_, Feb 06 2018 %Y A299252 Cf. A008579, A298802, A298805. %K A299252 nonn,easy,new %O A299252 0,2 %A A299252 _John Cannon_ and _N. J. A. Sloane_, Feb 06 2018 %I A298812 %S A298812 1,3,4,6,8,12,16,24,32,48,62,87,114,165,216,312,408,588,766,1104,1444, %T A298812 2082,2720,3921,5122,7383,9642,13902,18164,26184,34204,49308,64412, %U A298812 92856,121298,174867,228438,329313,430188,620160,810132,1167888,1525642,2199372,2873104,4141866,5410628,7799973,10189318,14688939 %N A298812 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^10 = 1 >. %C A298812 The initial coefficients for the group S, T : S^2 = T^3 = (S*T)^m = 1 > are converging to A029744 as m increases. %H A298812 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298812 Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,2,0,4,0,2,0,1,0,0,0,-1). %F A298812 G.f.: (-2*x^18 + 3*x^16 + 3*x^15 + 6*x^14 + 6*x^13 + 9*x^12 + 9*x^11 + 12*x^10 + 12*x^9 + 12*x^8 + 12*x^7 + 10*x^6 + 9*x^5 + 7*x^4 + 6*x^3 + 4*x^2 + 3*x + 1)/(x^16 - x^12 - 2*x^10 - 4*x^8 - 2*x^6 - x^4 + 1). %F A298812 a(n) = a(n-4) + 2*a(n-6) + 4*a(n-8) + 2*a(n-10) + a(n-12) - a(n-16) for n>16. - _Colin Barker_, Feb 06 2018 %o A298812 (MAGMA) See Magma program in A298805. %o A298812 (PARI) Vec((1 + 3*x + 4*x^2 + 6*x^3 + 7*x^4 + 9*x^5 + 10*x^6 + 12*x^7 + 12*x^8 + 12*x^9 + 12*x^10 + 9*x^11 + 9*x^12 + 6*x^13 + 6*x^14 + 3*x^15 + 3*x^16 - 2*x^18) / ((1 + x^2)^2*(1 + x^4)*(1 - 2*x^2 + x^4 - 2*x^6 + x^8)) + O(x^60)) \\ _Colin Barker_, Feb 06 2018 %Y A298812 Cf. A008579, A298802, A298805, A299252, A029744. %K A298812 nonn,easy,new %O A298812 0,2 %A A298812 _John Cannon_ and _N. J. A. Sloane_, Feb 06 2018 %I A298811 %S A298811 1,3,4,6,8,12,16,24,32,46,56,82,104,152,192,280,350,507,642,933,1176, %T A298811 1708,2152,3122,3940,5726,7216,10480,13212,19188,24190,35140,44300, %U A298811 64338,81112,117809,148522,215717,271960,394998,497972,723268,911828,1324360,1669626,2425008,3057212,4440362,5597988,8130648 %N A298811 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^9 = 1 >. %H A298811 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298811 Index entries for linear recurrences with constant coefficients, signature (-1,0,0,1,1,2,3,2,1,2,1,2,1,2,3,2,1,1,0,0,-1,-1). %F A298811 G.f.: (-2*x^24 - 2*x^23 + 3*x^22 + 6*x^21 + 9*x^20 + 12*x^19 + 15*x^18 + 18*x^17 + 21*x^16 + 25*x^15 + 27*x^14 + 27*x^13 + 27*x^12 + 27*x^11 + 27*x^10 + 27*x^9 + 23*x^8 + 21*x^7 + 19*x^6 + 16*x^5 + 13*x^4 + 10*x^3 + 7*x^2 + 4*x + 1)/(x^22 + x^21 - x^18 - x^17 - 2*x^16 - 3*x^15 - 2*x^14 - x^13 - 2*x^12 - x^11 - 2*x^10 - x^9 - 2*x^8 - 3*x^7 - 2*x^6 - x^5 - x^4 + x + 1). %F A298811 a(n) = -a(n-1) + a(n-4) + a(n-5) + 2*a(n-6) + 3*a(n-7) + 2*a(n-8) + a(n-9) + 2*a(n-10) + a(n-11) + 2*a(n-12) + a(n-13) + 2*a(n-14) + 3*a(n-15) + 2*a(n-16) + a(n-17) + a(n-18) - a(n-21) - a(n-22) for n>24. - _Colin Barker_, Feb 06 2018 %o A298811 (MAGMA) See Magma program in A298805. %o A298811 (PARI) Vec((1 + 4*x + 7*x^2 + 10*x^3 + 13*x^4 + 16*x^5 + 19*x^6 + 21*x^7 + 23*x^8 + 27*x^9 + 27*x^10 + 27*x^11 + 27*x^12 + 27*x^13 + 27*x^14 + 25*x^15 + 21*x^16 + 18*x^17 + 15*x^18 + 12*x^19 + 9*x^20 + 6*x^21 + 3*x^22 - 2*x^23 - 2*x^24) / ((1 - x + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)*(1 - 2*x^2 + x^6 - 2*x^10 + x^12)) + O(x^60)) \\ _Colin Barker_, Feb 06 2018 %Y A298811 Cf. A008579, A298802, A298805. %K A298811 nonn,easy,new %O A298811 0,2 %A A298811 _John Cannon_ and _N. J. A. Sloane_, Feb 06 2018 %I A298810 %S A298810 1,3,4,6,8,12,16,24,30,39,50,69,88,120,150,204,260,354,448,609,768, %T A298810 1047,1328,1806,2284,3108,3930,5352,6776,9219,11662,15873,20082,27336, %U A298810 34592,47076,59560,81066,102570,139605,176642,240411,304180,414006,523830 %N A298810 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^8 = 1 >. %H A298810 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298810 Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,3,0,1,0,0,0,-1). %F A298810 G.f.: (-2*x^14 + 3*x^12 + 3*x^11 + 6*x^10 + 6*x^9 + 9*x^8 + 9*x^7 + 9*x^6 + 9*x^5 + 7*x^4 + 6*x^3 + 4*x^2 + 3*x + 1)/(x^12 - x^8 - 3*x^6 - x^4 +1). %F A298810 a(n) = a(n-4) + 3*a(n-6) + a(n-8) - a(n-12) for n>12. - _Colin Barker_, Feb 06 2018 %o A298810 (MAGMA) See Magma program in A298805. %o A298810 (PARI) Vec((1 + 3*x + 4*x^2 + 6*x^3 + 7*x^4 + 9*x^5 + 9*x^6 + 9*x^7 + 9*x^8 + 6*x^9 + 6*x^10 + 3*x^11 + 3*x^12 - 2*x^14) / ((1 - x + x^2)*(1 + x + x^2)*(1 - x^2 - x^4 - x^6 + x^8)) + O(x^60)) \\ _Colin Barker_, Feb 06 2018 %Y A298810 Cf. A008579, A298802, A298805. %K A298810 nonn,easy,new %O A298810 0,2 %A A298810 _John Cannon_ and _N. J. A. Sloane_, Feb 06 2018 %I A298809 %S A298809 1,3,4,6,8,10,8,10,6,3,1 %N A298809 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^5 = 1 >. %C A298809 This group is finite, so the growth series is a polynomial. %o A298809 (MAGMA) See Magma program in A298805. %Y A298809 Cf. A008579, A298802, A298805. %K A298809 nonn,easy,fini,full,new %O A298809 0,2 %A A298809 _John Cannon_ and _N. J. A. Sloane_, Feb 06 2018 %I A298808 %S A298808 1,3,4,6,6,3,1 %N A298808 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^4 = 1 >. %C A298808 This group is finite, so the growth series is a polynomial. %o A298808 (MAGMA) See Magma program in A298805. %Y A298808 Cf. A008579, A298802, A298805. %K A298808 nonn,easy,fini,full,new %O A298808 0,2 %A A298808 _John Cannon_ and _N. J. A. Sloane_, Feb 06 2018 %I A299212 %S A299212 1,1,0,-2,-5,-4,4,21,35,23,-47,-165,-239,-78,479,1273,1508,-138,-4429, %T A299212 -9451,-8845,6207,37937,67123,45144,-83355,-308078,-455109,-166872, %U A299212 873799,2393041,2916869,-73472,-8133572,-17828640,-17294146,10383571,70275162,127401305,90368779,-147825714 %N A299212 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k). %H A299212 N. J. A. Sloane, Transforms %F A299212 G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k). %F A299212 a(0) = 1; a(n) = Sum_{k=1..n} A255528(k-1)*a(n-k). %t A299212 nmax = 40; CoefficientList[Series[1/(1 - x Product[1/(1 + x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x] %Y A299212 Antidiagonal sums of A279928. %Y A299212 Cf. A067687, A255528, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299211. %K A299212 sign,new %O A299212 0,4 %A A299212 _Ilya Gutkovskiy_, Feb 05 2018 %I A299211 %S A299211 1,1,0,-3,-6,-4,12,39,52,-9,-186,-392,-285,610,2291,3200,-150,-10626, %T A299211 -23487,-18841,32957,134848,198246,13961,-605248,-1409604,-1234474, %U A299211 1744213,7898753,12209679,2161666,-34344627,-84393284,-79993042,90692470,461463974,749309529,207447895,-1939084232 %N A299211 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)^k). %H A299211 N. J. A. Sloane, Transforms %F A299211 G.f.: 1/(1 - x*Product_{k>=1} (1 - x^k)^k). %F A299211 a(0) = 1; a(n) = Sum_{k=1..n} A073592(k-1)*a(n-k). %t A299211 nmax = 38; CoefficientList[Series[1/(1 - x Product[(1 - x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x] %Y A299211 Antidiagonal sums of A276554. %Y A299211 Cf. A067687, A073592, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299212. %K A299211 sign,new %O A299211 0,4 %A A299211 _Ilya Gutkovskiy_, Feb 05 2018 %I A299210 %S A299210 1,1,0,-2,-5,-3,5,20,27,17,-53,-152,-192,31,576,1110,694,-1297,-4519, %T A299210 -6160,-1107,13665,31914,30643,-19339,-119260,-196142,-103318,289543, %U A299210 859631,1062684,13710,-2690348,-5675946,-4940757,4167527,21343918,33874107,16524162,-51704908,-150454546 %N A299210 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + k*x^k)). %H A299210 N. J. A. Sloane, Transforms %F A299210 G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + k*x^k)). %F A299210 a(0) = 1; a(n) = Sum_{k=1..n} A022693(k-1)*a(n-k). %t A299210 nmax = 40; CoefficientList[Series[1/(1 - x Product[1/(1 + k x^k), {k, 1, nmax}]), {x, 0, nmax}], x] %Y A299210 Antidiagonal sums of A297325. %Y A299210 Cf. A022693, A067687, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299211, A299212. %K A299210 sign,new %O A299210 0,4 %A A299210 _Ilya Gutkovskiy_, Feb 05 2018 %I A299209 %S A299209 1,1,0,-3,-6,-5,11,37,59,13,-155,-402,-415,263,1981,3748,2289,-6643, %T A299209 -22642,-31322,-187,99040,229410,216823,-230029,-1223267,-2097812, %U A299209 -955237,4468902,13393758,16752461,-3891704,-62382597,-131974181,-106680562,173622424,741553622,1163057561,329176545 %N A299209 Expansion of 1/(1 - x*Product_{k>=1} (1 - k*x^k)). %H A299209 N. J. A. Sloane, Transforms %F A299209 G.f.: 1/(1 - x*Product_{k>=1} (1 - k*x^k)). %F A299209 a(0) = 1; a(n) = Sum_{k=1..n} A022661(k-1)*a(n-k). %t A299209 nmax = 38; CoefficientList[Series[1/(1 - x Product[1 - k x^k, {k, 1, nmax}]), {x, 0, nmax}], x] %Y A299209 Antidiagonal sums of A297323. %Y A299209 Cf. A022661, A067687, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299210, A299211, A299212. %K A299209 sign,new %O A299209 0,4 %A A299209 _Ilya Gutkovskiy_, Feb 05 2018 %I A299208 %S A299208 1,1,0,-1,-2,-1,1,3,3,1,-3,-6,-5,1,9,12,5,-9,-20,-18,1,26,38,21,-21, %T A299208 -61,-62,-9,72,120,81,-44,-177,-205,-64,186,366,293,-63,-496,-657, %U A299208 -304,445,1084,1014,33,-1341,-2053,-1238,959,3132,3378,770,-3474,-6260,-4619,1656,8809,10929,4306,-8520 %N A299208 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)). %H A299208 N. J. A. Sloane, Transforms %F A299208 G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + x^k)). %F A299208 a(0) = 1; a(n) = Sum_{k=1..n} A081362(k-1)*a(n-k). %t A299208 nmax = 60; CoefficientList[Series[1/(1 - x Product[1/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x] %Y A299208 Antidiagonal sums of A286352. %Y A299208 Cf. A067687, A081362, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299209, A299210, A299211, A299212. %K A299208 sign,new %O A299208 0,5 %A A299208 _Ilya Gutkovskiy_, Feb 05 2018 %I A299206 %S A299206 1,4830,-25499225,-359001100500,-488895969711875,15167983076643206250, %T A299206 97133231781274332671875,-271470664160664028370625000, %U A299206 -6054036890966043032024015234375,-15985594659896064584391569753906250,218396847859403327980436336954599609375 %N A299206 Ramanujan's tau function (or tau numbers (A000594)) for 5^n. %H A299206 Eric Weisstein's World of Mathematics, Tau Function %F A299206 G.f.: 1/(1-4830*x+48828125*x^2). %F A299206 a(n) = A000594(A000351(n)). - _Michel Marcus_, Feb 05 2018 %o A299206 (PARI) {a(n) = ramanujantau(5^n)} %Y A299206 Cf. A000594, A000351, A035174, A064556. %K A299206 sign,new %O A299206 0,2 %A A299206 _Seiichi Manyama_, Feb 05 2018 %I A299204 %S A299204 0,0,1,2,2,2,4,5,0,2,9,2,0,0,1,2,3,2,2,12,18,10,22,7,12,22,2,2,5,2,11, %T A299204 16,15,2,31,2,12,32,3,2,8,2,27,42,27,22,9,9,16,32,32,10,33,18,0,0,30, %U A299204 0,29,2,38,50,28,20,39,26,48,48,0,2,4,2,5,26,35,12 %N A299204 a(n) = A000594(n) mod (n-1). %o A299204 (PARI) {a(n) = ramanujantau(n)%(n-1)} %Y A299204 Cf. A000594, A273650, A299163, A299172, A299205. %K A299204 nonn,new %O A299204 2,4 %A A299204 _Seiichi Manyama_, Feb 05 2018 %I A297815 %S A297815 9,1,6,12,40,30,84,224,144,45,605,495,1170,1092,210,240,2448,4896, %T A297815 15846,3420,1750,462,15939,0,8100,67925,80730,19656,11774,164430,930, %U A297815 29760,197472,0,0,1260,23976,50616,54834,395200,1248860,4253340,75852,0,42570 %N A297815 Number of positive integers with n digits whose digit sum is equal to its digit product. %e A297815 The only term with two digits is 22: 2 * 2 = 2 + 2. %t A297815 cperm[w_] := Length[w]!/Times @@ ((Last /@ Tally[w])!); ric[s_, p_, w_, tg_] := Block[{d}, If[tg == 0, If[s == p, tot += cperm@ w], Do[ If[p*d > s + d + (tg-1)*9, Break[]]; ric[s+d, p*d, Append[w,d], tg-1], {d, Last@ w, 9}]]]; a[n_] := (tot=0; ric[#, #, {#}, n-1] & /@ Range[9]; tot); Array[a, 45] (* _Giovanni Resta_, Feb 05 2018 *) %o A297815 (PYTHON) %o A297815 import math %o A297815 def digitProd(natNumber): %o A297815 ...numberString = str(natNumber) %o A297815 ...digitProd = 1 %o A297815 ...for letter in numberString: %o A297815 ......digitProd = digitProd*int(letter) %o A297815 ...return digitProd %o A297815 def digitSum(natNumber): %o A297815 ...numberString = str(natNumber) %o A297815 ...digitSum = 0 %o A297815 ...for letter in numberString: %o A297815 ......digitSum = digitSum * int(letter) %o A297815 ...return digitSum %o A297815 for n in range(0, 24): %o A297815 ...count = 0 %o A297815 ...for a in range(int(math.pow(10,n)), int(math.pow(10, n+1))): %o A297815 ......if abs(digitProd(a) - digitSum(a)) == 0: %o A297815 .........count = count + 1 %o A297815 ...print(n+1, count) %Y A297815 Cf. A034710, A061672. %K A297815 nonn,base,new %O A297815 1,1 %A A297815 _Reiner Moewald_, Jan 06 2018 %E A297815 a(10) and a(23) corrected by and a(25)-a(45) from _Giovanni Resta_, Feb 05 2018 %I A298940 %S A298940 1,3,10,39,60,121,0,117,4920,0,0,0,28322,0,1434890,0,0,0,116226146,0, %T A298940 0,15690529803,0,108443565,66891206007,0,0,0,0,0,0,0,0,0,0,0, %U A298940 22514195294549868,0,405255515301897626,0,1823649818858539320,0,0,5861731560616733529,0,0,0 %N A298940 a(n) is the smallest positive integer k such that 3^n - 2 divides 3^(n + k) + 2, or 0 if there is no such k. %C A298940 3^n - 2 divides 3^(n + (2m + 1) * a(n)) + 2 for all nonnegative integers m. %C A298940 a(n) is the least positive integer k, if any, such that 3^k == -1 (mod 3^n-2). If the order of 3 mod p is odd for some prime p dividing 3^n-2, a(n)=0. - _Robert Israel_, Feb 05 2018 %H A298940 Robert Israel, Table of n, a(n) for n = 1..166 %e A298940 a(2) = 3 because 3^2 - 2 divides 3^5 + 2 and 3^2 - 2 does not divide any 3^x - 2 for 2 < x < 5. %e A298940 a(5) = 60 because 3^5 - 2 divides 3^65 + 2 and 3^5 - 2 does not divide any 3^x - 2 for 5 < x < 65. %p A298940 # This requires Maple 2016 or later %p A298940 f:= proc(n) local m,ps,a,p,q,phiq,v,br,ar; %p A298940 m:= 3^n-2; %p A298940 ps:= ifactors(m)[2]; %p A298940 a:= 0; %p A298940 for p in ps do %p A298940 q:= p[1]^p[2]; %p A298940 phiq:= (p[1]-1)*p[1]^(p[2]-1); %p A298940 v:= NumberTheory:-MultiplicativeOrder(3,q); %p A298940 if v::odd then return 0 fi; %p A298940 if p[2]=1 then br:= v/2 %p A298940 else br:= traperror(NumberTheory:-ModularLog(-1,3,q)); %p A298940 if br = lasterror then return 0 fi; %p A298940 fi; %p A298940 if a = 0 then a:= v; ar:= br %p A298940 else %p A298940 ar:= NumberTheory:-ChineseRemainder([ar,br],[a,v]); %p A298940 if ar = FAIL then return 0 fi; %p A298940 a:= ilcm(a, v); %p A298940 fi %p A298940 od: %p A298940 ar; %p A298940 end proc: %p A298940 f(1):= 1: %p A298940 map(f, [$1..50]); # _Robert Israel_, Feb 06 2018 %t A298940 a[1] = 1; a[n_] := If[IntegerQ[order = MultiplicativeOrder[3, 3^n - 2, {-1}]], order, 0]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 20}] (* _Jean-François Alcover_, Feb 06 2018, after _Robert Israel_ *) %o A298940 (Python) %o A298940 from sympy import discrete_log %o A298940 def A298940(n): %o A298940 if n == 1: %o A298940 return 1 %o A298940 try: %o A298940 return discrete_log(3**n-2,-1,3) %o A298940 except ValueError: %o A298940 return 0 # _Chai Wah Wu_, Feb 05 2018 %o A298940 (PARI) a(n) = if(n==1, return(1)); my(l = znlog(-1, Mod(3, 3^n - 2))); if(l == [], return(0), return(l)) \\ _Iain Fox_, Feb 06 2018 %Y A298940 Cf. A168607, A298827. %K A298940 nonn,new %O A298940 1,2 %A A298940 _Luke W. Richards_, Jan 29 2018 %E A298940 Corrected by _Robert Israel_, Feb 05 2018 %I A299249 %S A299249 1,1,1,1,5,1,1,7,7,1,1,18,6,18,1,1,31,18,18,31,1,1,65,30,55,30,65,1,1, %T A299249 130,87,192,192,87,130,1,1,253,202,652,1095,652,202,253,1,1,519,526, %U A299249 2002,5042,5042,2002,526,519,1,1,1018,1449,6741,21251,35320,21251,6741,1449 %N A299249 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299249 Table starts %C A299249 .1...1....1.....1......1........1.........1..........1............1 %C A299249 .1...5....7....18.....31.......65.......130........253..........519 %C A299249 .1...7....6....18.....30.......87.......202........526.........1449 %C A299249 .1..18...18....55....192......652......2002.......6741........23631 %C A299249 .1..31...30...192...1095.....5042.....21251.....111818.......577544 %C A299249 .1..65...87...652...5042....35320....223750....1634125.....12440063 %C A299249 .1.130..202..2002..21251...223750...2044405...21212789....227105938 %C A299249 .1.253..526..6741.111818..1634125..21212789..321811361...4948662848 %C A299249 .1.519.1449.23631.577544.12440063.227105938.4948662848.111378069579 %H A299249 R. H. Hardin, Table of n, a(n) for n = 1..197 %F A299249 Empirical for column k: %F A299249 k=1: a(n) = a(n-1) %F A299249 k=2: a(n) = a(n-1) +3*a(n-2) -4*a(n-4) for n>5 %F A299249 k=3: [order 19] for n>20 %F A299249 k=4: [order 72] for n>73 %e A299249 Some solutions for n=5 k=4 %e A299249 ..0..0..1..1. .0..1..0..0. .0..0..0..0. .0..0..1..1. .0..1..0..1 %e A299249 ..0..1..0..1. .0..0..0..1. .0..0..0..0. .1..0..1..0. .1..1..1..1 %e A299249 ..1..1..0..0. .0..1..1..1. .1..1..0..0. .1..1..1..1. .0..1..1..0 %e A299249 ..0..0..1..1. .1..0..0..0. .1..1..0..0. .0..1..1..0. .1..1..0..0 %e A299249 ..1..0..1..0. .1..1..0..1. .1..1..0..0. .1..1..0..0. .0..1..1..0 %Y A299249 Column 2 is A297937. %K A299249 nonn,tabl,new %O A299249 1,5 %A A299249 _R. H. Hardin_, Feb 05 2018 %I A299248 %S A299248 1,130,202,2002,21251,223750,2044405,21212789,227105938,2271071651, %T A299248 23645777755,246352885297,2552827451230,26493876070013, %U A299248 275607234143594,2866930315549461,29800020165962133,310027901670675161 %N A299248 Number of nX7 0..1 arrays with every element equal to 0, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299248 Column 7 of A299249. %H A299248 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299248 Some solutions for n=5 %e A299248 ..0..0..1..1..0..1..0. .0..1..1..0..1..0..0. .0..0..1..0..1..0..0 %e A299248 ..1..0..0..1..1..1..1. .1..1..0..0..0..0..1. .0..1..1..1..1..1..0 %e A299248 ..1..1..1..1..1..1..0. .0..0..0..0..0..0..0. .1..1..1..0..0..0..1 %e A299248 ..1..0..1..1..1..0..0. .0..1..0..1..1..0..1. .0..0..1..0..1..1..1 %e A299248 ..0..0..1..0..1..1..0. .1..1..0..0..1..0..0. .1..0..0..1..1..0..1 %Y A299248 Cf. A299249. %K A299248 nonn,new %O A299248 1,2 %A A299248 _R. H. Hardin_, Feb 05 2018 %I A299247 %S A299247 1,65,87,652,5042,35320,223750,1634125,12440063,85743693,626402775, %T A299247 4593707989,33301518925,242065110566,1767254828555,12896269372285, %U A299247 93959680164405,685715068810035,5004838983561794,36516480551676091 %N A299247 Number of nX6 0..1 arrays with every element equal to 0, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299247 Column 6 of A299249. %H A299247 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299247 Some solutions for n=5 %e A299247 ..0..0..1..1..1..0. .0..1..1..1..0..0. .0..1..1..0..1..1. .0..1..0..1..1..0 %e A299247 ..1..0..0..1..0..0. .1..1..0..0..1..0. .0..0..1..1..1..0. .0..0..0..0..1..1 %e A299247 ..1..1..1..1..1..0. .1..0..1..0..1..1. .0..1..1..1..1..1. .1..0..0..1..1..0 %e A299247 ..1..0..1..1..1..1. .1..0..1..0..1..0. .1..1..1..1..0..1. .1..1..0..0..1..1 %e A299247 ..0..0..1..0..1..0. .1..1..0..1..1..1. .1..0..1..0..0..0. .1..0..0..1..1..0 %Y A299247 Cf. A299249. %K A299247 nonn,new %O A299247 1,2 %A A299247 _R. H. Hardin_, Feb 05 2018 %I A299246 %S A299246 1,31,30,192,1095,5042,21251,111818,577544,2671192,13666341,68251730, %T A299246 337908406,1685990324,8431949244,42087273799,209909970220, %U A299246 1049387080384,5241184058073,26176899422346,130786862915157,653424242725917 %N A299246 Number of nX5 0..1 arrays with every element equal to 0, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299246 Column 5 of A299249. %H A299246 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299246 Some solutions for n=5 %e A299246 ..0..0..1..0..0. .0..0..1..0..0. .0..1..0..1..1. .0..1..0..1..1 %e A299246 ..1..0..0..0..1. .0..1..1..1..0. .1..1..0..0..1. .0..0..0..0..1 %e A299246 ..0..0..0..0..0. .1..1..1..1..1. .1..0..0..0..0. .1..1..0..0..0 %e A299246 ..1..1..0..0..1. .0..0..1..1..0. .0..0..0..1..0. .0..1..1..0..1 %e A299246 ..1..0..0..1..1. .0..1..1..0..0. .1..0..1..1..1. .1..1..0..0..0 %Y A299246 Cf. A299249. %K A299246 nonn,new %O A299246 1,2 %A A299246 _R. H. Hardin_, Feb 05 2018 %I A299245 %S A299245 1,18,18,55,192,652,2002,6741,23631,79836,274822,954282,3306096, %T A299245 11462662,39849725,138545796,481560761,1674865229,5826326576, %U A299245 20267377567,70508970351,245315398653,853512851244,2969629901018,10332484940737 %N A299245 Number of nX4 0..1 arrays with every element equal to 0, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299245 Column 4 of A299249. %H A299245 R. H. Hardin, Table of n, a(n) for n = 1..210 %H A299245 R. H. Hardin, Empirical recurrence of order 72 %F A299245 Empirical recurrence of order 72 (see link above) %e A299245 Some solutions for n=5 %e A299245 ..0..1..1..0. .0..0..1..1. .0..0..1..1. .0..1..1..1. .0..1..1..0 %e A299245 ..0..0..1..1. .0..0..1..1. .1..0..1..0. .0..0..1..0. .0..0..1..1 %e A299245 ..1..1..1..0. .0..0..1..1. .1..1..0..0. .0..1..1..1. .1..1..0..1 %e A299245 ..0..1..1..1. .1..1..1..1. .0..1..1..1. .1..1..1..0. .0..1..0..0 %e A299245 ..1..1..0..1. .1..1..1..1. .1..1..0..1. .1..0..1..1. .1..1..1..0 %Y A299245 Cf. A299249. %K A299245 nonn,new %O A299245 1,2 %A A299245 _R. H. Hardin_, Feb 05 2018 %I A299244 %S A299244 1,7,6,18,30,87,202,526,1449,3893,10886,30529,85878,243545,691293, %T A299244 1966629,5603311,15974714,45572960,130060050,371260631,1059964088, %U A299244 3026563164,8642492229,24680273281,70481398248,201283524625,574841555052 %N A299244 Number of nX3 0..1 arrays with every element equal to 0, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299244 Column 3 of A299249. %H A299244 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299244 Empirical: a(n) = 4*a(n-1) -9*a(n-3) -10*a(n-4) +24*a(n-5) +14*a(n-6) -19*a(n-7) -15*a(n-8) -4*a(n-9) -32*a(n-10) +54*a(n-11) +126*a(n-12) -72*a(n-13) -96*a(n-14) -14*a(n-15) -10*a(n-16) +32*a(n-17) +24*a(n-18) +4*a(n-19) for n>20 %e A299244 Some solutions for n=10 %e A299244 ..0..1..0. .0..0..1. .0..1..1. .0..1..1. .0..0..1. .0..0..1. .0..1..1 %e A299244 ..0..0..0. .1..0..0. .1..1..0. .1..1..0. .1..0..0. .1..0..0. .1..1..0 %e A299244 ..1..1..0. .0..0..1. .0..0..0. .0..0..0. .1..1..1. .0..0..1. .0..1..1 %e A299244 ..0..1..0. .1..1..1. .0..0..1. .0..1..0. .0..1..1. .1..0..0. .0..0..0 %e A299244 ..0..0..1. .1..1..0. .0..0..0. .1..1..0. .1..1..1. .0..0..1. .1..0..0 %e A299244 ..1..1..1. .1..1..1. .1..1..0. .0..0..0. .1..0..0. .1..1..1. .0..0..0 %e A299244 ..1..1..0. .1..1..0. .1..1..0. .0..0..1. .1..0..1. .1..1..0. .0..1..1 %e A299244 ..1..1..1. .1..1..1. .0..0..0. .0..0..0. .0..1..1. .1..1..1. .1..1..0 %e A299244 ..0..0..1. .0..0..1. .0..1..0. .1..1..0. .0..0..0. .0..0..1. .0..0..0 %e A299244 ..1..0..0. .1..0..0. .1..1..1. .0..1..1. .0..1..0. .1..0..0. .0..1..0 %Y A299244 Cf. A299249. %K A299244 nonn,new %O A299244 1,2 %A A299244 _R. H. Hardin_, Feb 05 2018 %I A299243 %S A299243 1,5,6,55,1095,35320,2044405,321811361,111378069579,67948097097737 %N A299243 Number of nXn 0..1 arrays with every element equal to 0, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299243 Diagonal of A299249. %e A299243 Some solutions for n=5 %e A299243 ..0..0..1..0..0. .0..1..1..1..1. .0..0..1..0..0. .0..0..0..0..0 %e A299243 ..0..1..1..0..1. .1..1..0..0..1. .0..1..1..1..0. .1..0..1..1..0 %e A299243 ..1..1..1..1..1. .1..0..1..0..1. .1..1..1..1..1. .0..0..1..0..1 %e A299243 ..0..1..1..1..0. .1..0..0..1..1. .0..0..1..0..1. .1..0..1..0..1 %e A299243 ..0..0..1..0..0. .1..1..1..1..0. .0..1..1..0..0. .0..0..0..1..1 %Y A299243 Cf. A299249. %K A299243 nonn,new %O A299243 1,2 %A A299243 _R. H. Hardin_, Feb 05 2018 %I A299228 %S A299228 0,1,1,1,4,1,2,17,17,2,3,61,113,61,3,5,216,628,628,216,5,8,793,3669, %T A299228 5663,3669,793,8,13,2907,21792,51862,51862,21792,2907,13,21,10622, %U A299228 128610,486305,766094,486305,128610,10622,21,34,38809,758715,4532025,11568483 %N A299228 T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299228 Table starts %C A299228 ..0.....1.......1.........2...........3.............5...............8 %C A299228 ..1.....4......17........61.........216...........793............2907 %C A299228 ..1....17.....113.......628........3669.........21792..........128610 %C A299228 ..2....61.....628......5663.......51862........486305.........4532025 %C A299228 ..3...216....3669.....51862......766094......11568483.......173234478 %C A299228 ..5...793...21792....486305....11568483.....281251424......6766315496 %C A299228 ..8..2907..128610...4532025...173234478....6766315496....260938102438 %C A299228 .13.10622..758715..42210111..2594065569..162902164724..10077166692402 %C A299228 .21.38809.4478515.393354513.38873368814.3925885203249.389637415718865 %H A299228 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299228 Empirical for column k: %F A299228 k=1: a(n) = a(n-1) +a(n-2) %F A299228 k=2: a(n) = 3*a(n-1) +a(n-2) +4*a(n-3) +4*a(n-4) for n>5 %F A299228 k=3: [order 14] for n>16 %F A299228 k=4: [order 49] for n>51 %e A299228 Some solutions for n=5 k=4 %e A299228 ..0..1..1..0. .0..1..0..0. .0..0..0..0. .0..1..1..1. .0..0..0..0 %e A299228 ..0..0..0..0. .0..1..1..1. .1..1..1..0. .0..0..0..1. .1..1..1..0 %e A299228 ..0..0..0..0. .0..1..1..1. .0..1..1..1. .0..0..0..1. .0..1..1..1 %e A299228 ..1..0..1..0. .0..1..0..1. .0..1..1..1. .0..1..0..1. .0..1..1..1 %e A299228 ..1..1..0..1. .1..0..0..1. .1..0..0..1. .0..1..1..0. .0..0..0..1 %Y A299228 Column 1 is A000045(n-1). %Y A299228 Column 2 is A297917. %K A299228 nonn,tabl,new %O A299228 1,5 %A A299228 _R. H. Hardin_, Feb 05 2018 %I A299227 %S A299227 8,2907,128610,4532025,173234478,6766315496,260938102438, %T A299227 10077166692402,389637415718865,15058710314116325,581977479263622822, %U A299227 22493737410496748938,869383064950398158845,33601436598811749671626 %N A299227 Number of nX7 0..1 arrays with every element equal to 1, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299227 Column 7 of A299228. %H A299227 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299227 Some solutions for n=5 %e A299227 ..0..0..0..1..1..1..0. .0..0..1..1..1..1..1. .0..0..1..0..1..1..0 %e A299227 ..0..1..0..0..1..1..0. .0..1..0..1..1..0..0. .0..1..0..0..1..1..0 %e A299227 ..0..1..0..1..1..1..1. .0..1..0..1..1..1..1. .0..1..0..1..1..1..1 %e A299227 ..0..0..0..1..1..0..1. .0..0..0..1..0..1..0. .0..0..0..1..1..1..1 %e A299227 ..0..0..1..0..0..0..0. .0..0..1..0..1..0..0. .0..0..1..1..1..0..0 %Y A299227 Cf. A299228. %K A299227 nonn,new %O A299227 1,1 %A A299227 _R. H. Hardin_, Feb 05 2018 %I A299226 %S A299226 5,793,21792,486305,11568483,281251424,6766315496,162902164724, %T A299226 3925885203249,94583364942194,2278655600670624,54899678166337106, %U A299226 1322688236196434142,31867127372632598869,767767745513800333670 %N A299226 Number of nX6 0..1 arrays with every element equal to 1, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299226 Column 6 of A299228. %H A299226 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299226 Some solutions for n=4 %e A299226 ..0..0..0..0..1..0. .0..0..0..0..1..0. .0..0..0..0..0..0. .0..0..0..0..0..1 %e A299226 ..0..1..1..0..1..0. .0..1..0..1..0..1. .0..1..1..1..1..1. .0..1..0..0..1..1 %e A299226 ..1..0..0..0..1..0. .1..0..1..1..0..1. .1..0..1..1..0..0. .0..1..0..0..0..1 %e A299226 ..1..1..1..0..1..0. .1..1..0..0..0..1. .1..1..1..0..1..1. .0..0..1..1..1..1 %Y A299226 Cf. A299228. %K A299226 nonn,new %O A299226 1,1 %A A299226 _R. H. Hardin_, Feb 05 2018 %I A299225 %S A299225 3,216,3669,51862,766094,11568483,173234478,2594065569,38873368814, %T A299225 582449031740,8726553150427,130750114509849,1959028588732256, %U A299225 29352000502032743,439779727081644491,6589202287611286020 %N A299225 Number of nX5 0..1 arrays with every element equal to 1, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299225 Column 5 of A299228. %H A299225 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299225 Some solutions for n=5 %e A299225 ..0..0..0..1..0. .0..0..0..0..1. .0..0..0..1..1. .0..0..0..1..1 %e A299225 ..0..1..0..1..0. .1..1..0..0..1. .0..1..0..0..1. .0..1..0..0..0 %e A299225 ..0..1..0..1..0. .1..0..0..0..0. .1..1..0..1..0. .0..1..1..1..1 %e A299225 ..0..1..0..1..0. .1..0..0..1..0. .0..0..1..0..1. .0..1..0..0..0 %e A299225 ..0..1..1..1..1. .1..1..1..1..1. .1..1..0..1..0. .0..0..1..1..1 %Y A299225 Cf. A299228. %K A299225 nonn,new %O A299225 1,1 %A A299225 _R. H. Hardin_, Feb 05 2018 %I A299224 %S A299224 2,61,628,5663,51862,486305,4532025,42210111,393354513,3665540260, %T A299224 34156287528,318279356683,2965838291898,27636672383375, %U A299224 257527745652155,2399729999704478,22361489149827987,208371854494502461 %N A299224 Number of nX4 0..1 arrays with every element equal to 1, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299224 Column 4 of A299228. %H A299224 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299224 Empirical: a(n) = 8*a(n-1) +10*a(n-2) +56*a(n-3) -203*a(n-4) -967*a(n-5) -1599*a(n-6) -306*a(n-7) +13357*a(n-8) +22549*a(n-9) +31178*a(n-10) -83309*a(n-11) -162226*a(n-12) -264159*a(n-13) +263535*a(n-14) +610742*a(n-15) +1107070*a(n-16) +51242*a(n-17) -80850*a(n-18) -556862*a(n-19) -37820*a(n-20) -3297808*a(n-21) -5672783*a(n-22) -6349408*a(n-23) +8730*a(n-24) +4479264*a(n-25) +6070656*a(n-26) +6316587*a(n-27) +11010346*a(n-28) +14031549*a(n-29) +7970488*a(n-30) -595394*a(n-31) -8486885*a(n-32) -8903264*a(n-33) -7863100*a(n-34) -16214548*a(n-35) -7990141*a(n-36) -5093572*a(n-37) -442625*a(n-38) +2985779*a(n-39) +2101405*a(n-40) +2884148*a(n-41) +435040*a(n-42) +998805*a(n-43) +634665*a(n-44) -37051*a(n-45) +2313*a(n-46) -57573*a(n-47) +10002*a(n-48) -124*a(n-49) for n>51 %e A299224 Some solutions for n=5 %e A299224 ..0..0..0..0. .0..1..1..0. .0..0..1..1. .0..1..0..0. .0..1..1..1 %e A299224 ..1..0..1..0. .0..0..0..1. .0..1..0..1. .0..1..0..1. .0..0..0..1 %e A299224 ..1..1..1..1. .0..0..0..1. .0..1..1..1. .0..0..0..1. .0..0..0..0 %e A299224 ..1..1..1..1. .1..0..0..0. .0..1..1..1. .0..0..0..1. .0..1..0..0 %e A299224 ..1..0..0..1. .0..1..0..0. .1..0..0..1. .0..1..1..0. .0..0..1..0 %Y A299224 Cf. A299228. %K A299224 nonn,new %O A299224 1,1 %A A299224 _R. H. Hardin_, Feb 05 2018 %I A299223 %S A299223 1,17,113,628,3669,21792,128610,758715,4478515,26434415,156022822, %T A299223 920895737,5435426140,32081622538,189355989729,1117639651783, %U A299223 6596666844249,38935638345812,229810595428761,1356415664939788 %N A299223 Number of nX3 0..1 arrays with every element equal to 1, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299223 Column 3 of A299228. %H A299223 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299223 Empirical: a(n) = 5*a(n-1) +3*a(n-2) +16*a(n-3) -7*a(n-4) -32*a(n-5) -25*a(n-6) -62*a(n-7) +44*a(n-8) -57*a(n-9) -12*a(n-10) -74*a(n-11) +46*a(n-12) -20*a(n-13) -16*a(n-14) for n>16 %e A299223 Some solutions for n=5 %e A299223 ..0..1..1. .0..0..1. .0..0..1. .0..1..1. .0..1..0. .0..0..1. .0..0..1 %e A299223 ..0..1..0. .1..1..1. .1..0..1. .0..0..0. .0..1..0. .1..1..1. .1..1..1 %e A299223 ..0..0..0. .1..1..1. .1..1..1. .0..0..0. .0..0..0. .1..1..1. .1..1..1 %e A299223 ..0..0..0. .1..0..1. .1..1..1. .0..1..0. .0..0..0. .1..0..1. .1..0..1 %e A299223 ..0..1..1. .0..0..0. .1..0..0. .1..1..0. .1..1..0. .0..1..0. .0..1..1 %Y A299223 Cf. A299228. %K A299223 nonn,new %O A299223 1,2 %A A299223 _R. H. Hardin_, Feb 05 2018 %I A299222 %S A299222 0,4,113,5663,766094,281251424,260938102438,624662873923282, %T A299222 3865535525302817693 %N A299222 Number of nXn 0..1 arrays with every element equal to 1, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299222 Diagonal of A299228. %e A299222 Some solutions for n=5 %e A299222 ..0..0..0..0..1. .0..0..1..0..0. .0..0..1..0..0. .0..0..0..1..0 %e A299222 ..0..1..0..0..1. .0..1..0..0..1. .0..1..0..0..1. .0..1..1..1..0 %e A299222 ..1..0..0..0..0. .1..1..1..0..1. .1..0..1..1..0. .1..0..0..0..1 %e A299222 ..0..1..1..1..1. .0..1..0..1..0. .0..1..0..0..1. .0..1..0..0..1 %e A299222 ..0..1..1..0..0. .1..0..0..0..1. .1..0..0..1..0. .0..0..0..0..0 %Y A299222 Cf. A299228. %K A299222 nonn,new %O A299222 1,2 %A A299222 _R. H. Hardin_, Feb 05 2018 %I A298807 %S A298807 1,4,8,16,32,64,126,242,472,920,1792,3486,6788,13216,25730,50092, %T A298807 97518,189860,369628,719612,1400980,2727504,5310068,10337932,20126468, %U A298807 39183340,76284330,148514636,289136638,562907480,1095899956,2133559698,4153734080,8086723216,15743687792,30650697262,59672502090 %N A298807 Growth series for group with presentation < S, T : S^3 = T^3 = (S*T)^6 = 1 >. %H A298807 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298807 Index entries for linear recurrences with constant coefficients, signature (0,1,2,3,5,3,2,1,0,-1). %F A298807 G.f.: (-2*x^11 + 3*x^10 + 6*x^9 + 9*x^8 + 12*x^7 + 15*x^6 + 15*x^5 + 13*x^4 + 10*x^3 + 7*x^2 + 4*x + 1)/(x^10 - x^8 - 2*x^7 - 3*x^6 - 5*x^5 - 3*x^4 - 2*x^3 - x^2 + 1). %F A298807 a(n) = a(n-2) + 2*a(n-3) + 3*a(n-4) + 5*a(n-5) + 3*a(n-6) + 2*a(n-7) + a(n-8) - a(n-10) for n>11. - _Colin Barker_, Feb 06 2018 %o A298807 (MAGMA) See Magma program in A298805. %o A298807 (PARI) Vec((1 + 4*x + 7*x^2 + 10*x^3 + 13*x^4 + 15*x^5 + 15*x^6 + 12*x^7 + 9*x^8 + 6*x^9 + 3*x^10 - 2*x^11) / ((1 + x + x^2 + x^3 + x^4)*(1 - x - x^2 - x^3 - x^4 - x^5 + x^6)) + O(x^40)) \\ _Colin Barker_, Feb 06 2018 %Y A298807 Cf. A008579, A298802, A298805. %K A298807 nonn,easy,new %O A298807 0,2 %A A298807 _John Cannon_ and _N. J. A. Sloane_, Feb 04 2018 %I A299221 %S A299221 1,1,1,1,5,1,1,12,12,1,1,37,22,37,1,1,104,81,81,104,1,1,301,307,427, %T A299221 307,301,1,1,864,1201,2338,2338,1201,864,1,1,2485,5066,13458,21730, %U A299221 13458,5066,2485,1,1,7144,21292,84948,202841,202841,84948,21292,7144,1,1,20541 %N A299221 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299221 Table starts %C A299221 .1....1.....1.......1.........1...........1............1..............1 %C A299221 .1....5....12......37.......104.........301..........864...........2485 %C A299221 .1...12....22......81.......307........1201.........5066..........21292 %C A299221 .1...37....81.....427......2338.......13458........84948.........543741 %C A299221 .1..104...307....2338.....21730......202841......1992466.......19685956 %C A299221 .1..301..1201...13458....202841.....3096833.....48911434......775504649 %C A299221 .1..864..5066...84948...1992466....48911434...1226106440....30729398000 %C A299221 .1.2485.21292..543741..19685956...775504649..30729398000..1213390065190 %C A299221 .1.7144.90443.3534493.195564094.12397088059.778116037031.48505143319432 %H A299221 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299221 Empirical for column k: %F A299221 k=1: a(n) = a(n-1) %F A299221 k=2: a(n) = 5*a(n-2) +8*a(n-3) +4*a(n-4) %F A299221 k=3: [order 19] for n>20 %F A299221 k=4: [order 66] for n>68 %e A299221 Some solutions for n=5 k=4 %e A299221 ..0..1..0..1. .0..0..1..0. .0..0..1..1. .0..0..0..0. .0..0..1..0 %e A299221 ..1..1..0..0. .1..0..0..0. .0..1..1..0. .0..1..0..1. .0..0..1..1 %e A299221 ..1..0..0..1. .0..0..0..1. .0..1..0..0. .1..1..1..1. .1..0..1..0 %e A299221 ..0..0..1..1. .0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..1..1 %e A299221 ..1..0..1..1. .0..1..0..0. .0..1..1..0. .1..0..0..1. .1..0..1..1 %Y A299221 Column 2 is A297909. %K A299221 nonn,tabl,new %O A299221 1,5 %A A299221 _R. H. Hardin_, Feb 05 2018 %I A299220 %S A299220 1,864,5066,84948,1992466,48911434,1226106440,30729398000, %T A299220 778116037031,19790046600537,502545421141263,12776486958379265, %U A299220 324936759791317709,8263214485959250239,210151973765681649219,5344770791861074550207 %N A299220 Number of nX7 0..1 arrays with every element equal to 0, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299220 Column 7 of A299221. %H A299220 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299220 Some solutions for n=5 %e A299220 ..0..0..0..1..0..0..0. .0..0..0..0..0..0..0. .0..0..0..1..0..0..0 %e A299220 ..0..0..1..0..1..1..0. .0..0..1..1..1..1..0. .0..0..1..1..0..0..1 %e A299220 ..0..1..0..1..0..1..1. .0..1..0..1..0..1..1. .0..1..0..0..0..0..0 %e A299220 ..0..0..1..1..0..1..0. .0..0..1..0..0..0..1. .0..1..0..1..1..1..1 %e A299220 ..0..0..0..0..0..1..1. .0..0..0..1..1..1..1. .0..0..1..1..1..0..1 %Y A299220 Cf. A299221. %K A299220 nonn,new %O A299220 1,2 %A A299220 _R. H. Hardin_, Feb 05 2018 %I A299219 %S A299219 1,301,1201,13458,202841,3096833,48911434,775504649,12397088059, %T A299219 198792990540,3185458609681,51097770758696,819822840723877, %U A299219 13152747682174642,211032523843679943,3386021726043218447 %N A299219 Number of nX6 0..1 arrays with every element equal to 0, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299219 Column 6 of A299221. %H A299219 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299219 Some solutions for n=5 %e A299219 ..0..0..1..0..0..1. .0..0..1..1..0..0. .0..0..0..0..1..1. .0..1..1..0..0..1 %e A299219 ..1..0..1..1..0..0. .0..0..0..1..1..0. .1..0..1..1..0..1. .1..1..1..0..1..1 %e A299219 ..0..0..1..0..0..0. .1..1..1..0..0..1. .1..1..0..0..1..1. .0..0..0..1..0..0 %e A299219 ..0..1..1..0..1..0. .0..1..1..0..1..0. .0..1..0..1..0..0. .0..1..0..1..0..0 %e A299219 ..0..0..0..0..1..1. .1..1..1..1..0..0. .0..0..1..1..0..1. .1..1..1..1..0..1 %Y A299219 Cf. A299221. %K A299219 nonn,new %O A299219 1,2 %A A299219 _R. H. Hardin_, Feb 05 2018 %I A299218 %S A299218 1,104,307,2338,21730,202841,1992466,19685956,195564094,1955287119, %T A299218 19542994060,195559255779,1957603231193,19597685801829, %U A299218 196216526062905,1964628425669218,19671255391318388,196964840936957656 %N A299218 Number of nX5 0..1 arrays with every element equal to 0, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299218 Column 5 of A299221. %H A299218 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299218 Some solutions for n=5 %e A299218 ..0..0..1..0..0. .0..0..1..1..1. .0..1..0..0..0. .0..0..1..0..0 %e A299218 ..0..1..1..0..0. .0..1..0..1..0. .1..1..1..0..0. .0..1..1..0..0 %e A299218 ..1..1..0..1..1. .1..1..0..0..0. .1..0..0..1..0. .1..0..1..1..1 %e A299218 ..0..1..0..1..0. .1..1..0..0..0. .1..0..0..1..1. .1..0..0..0..1 %e A299218 ..0..0..0..0..0. .1..0..0..0..1. .1..1..0..1..1. .1..1..0..1..1 %Y A299218 Cf. A299221. %K A299218 nonn,new %O A299218 1,2 %A A299218 _R. H. Hardin_, Feb 05 2018 %I A295874 %S A295874 7,2,6,5,6,4,1,9,3,2,7,4,0,4,3,6,2,6,4,4,1,6,2,4,1,3,0,1,0,1,1,3,3,4, %T A295874 1,5,5,0,4,3,3,0,8,4,7,2,3,9,1,2,0,0,2,2,4,2,0,2,8,4,1,0,3,4,6,4,5,4, %U A295874 3,1,7,4,8,1,3,3,2,2,0,8,1,3,2,2,2,0,2,4,6,5,7,6,3,4,1,0,2,0,7,9,6,3,4,0,5,5,6 %N A295874 Decimal expansion of the real positive fixed point of the Dirichlet beta function. %H A295874 Eric Weisstein's MathWorld, Dirichlet Beta Function. %H A295874 Wikipedia, Dirichlet beta function. %e A295874 0.72656419327404362644162413010113341550433084723912002242028410346454317481... %p A295874 Digits:= 140: %p A295874 f:= s-> sum((-1)^n/(2*n+1)^s, n=0..infinity): %p A295874 fsolve(f(x)=x, x); # _Alois P. Heinz_, Feb 05 2018 %t A295874 RealDigits[ FindRoot[ DirichletBeta[x] == x, {x, 0}, WorkingPrecision -> 2^7, AccuracyGoal -> 2^8, PrecisionGoal -> 2^7][[1, 2]], 10, 111][[1]] (* _Robert G. Wilson v_, Jan 07 2018 *) %o A295874 (PARI) solve(x=0,1,sumalt(n=0,((-1)^n)/(2*n+1)^x)-x) %Y A295874 Cf. A261624. %K A295874 nonn,cons,new %O A295874 0,1 %A A295874 _Michal Paulovic_, Dec 31 2017 %I A299217 %S A299217 1,37,81,427,2338,13458,84948,543741,3534493,23192676,152386263, %T A299217 1003140993,6609135301,43551719814,287046844098,1892053858627, %U A299217 12471668276710,82210001337166,541910867484011,3572172049448373 %N A299217 Number of nX4 0..1 arrays with every element equal to 0, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299217 Column 4 of A299221. %H A299217 R. H. Hardin, Table of n, a(n) for n = 1..210 %H A299217 R. H. Hardin, Empirical recurrence of order 66 %F A299217 Empirical recurrence of order 66 (see link above) %e A299217 Some solutions for n=5 %e A299217 ..0..1..1..1. .0..1..1..0. .0..1..0..1. .0..0..0..1. .0..1..1..0 %e A299217 ..0..0..1..0. .1..1..0..0. .1..1..0..0. .1..0..1..1. .1..1..1..1 %e A299217 ..0..1..1..1. .1..0..0..1. .1..1..0..0. .1..1..0..1. .1..1..1..0 %e A299217 ..0..1..0..0. .1..0..0..0. .0..0..1..1. .0..1..0..1. .0..0..1..1 %e A299217 ..0..0..0..1. .1..1..0..0. .0..0..1..0. .0..0..1..1. .0..1..1..1 %Y A299217 Cf. A299221. %K A299217 nonn,new %O A299217 1,2 %A A299217 _R. H. Hardin_, Feb 05 2018 %I A299216 %S A299216 1,12,22,81,307,1201,5066,21292,90443,387999,1664166,7150000,30748156, %T A299216 132241210,568885960,2447521291,10530203290,45306613634,194935607759, %U A299216 838730766281,3608741827447,15527073831286,66807269901814 %N A299216 Number of nX3 0..1 arrays with every element equal to 0, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299216 Column 3 of A299221. %H A299216 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299216 Empirical: a(n) = 3*a(n-1) +7*a(n-2) +2*a(n-3) -27*a(n-4) -27*a(n-5) +3*a(n-6) -101*a(n-7) -69*a(n-8) +201*a(n-9) +105*a(n-10) -137*a(n-11) -32*a(n-12) +47*a(n-13) +21*a(n-14) -17*a(n-15) +37*a(n-16) +33*a(n-17) -12*a(n-18) -4*a(n-19) for n>20 %e A299216 Some solutions for n=5 %e A299216 ..0..0..1. .0..1..0. .0..0..1. .0..1..0. .0..1..1. .0..0..1. .0..1..0 %e A299216 ..0..1..1. .0..0..0. .0..1..1. .0..0..0. .0..0..1. .0..0..0. .1..1..1 %e A299216 ..0..0..0. .0..0..0. .0..0..0. .0..0..1. .1..1..1. .1..0..0. .0..0..1 %e A299216 ..1..0..1. .1..1..0. .0..0..0. .1..1..1. .0..1..0. .0..0..1. .1..0..1 %e A299216 ..1..1..1. .1..0..0. .1..0..1. .0..1..0. .1..1..1. .0..0..0. .1..1..1 %Y A299216 Cf. A299221. %K A299216 nonn,new %O A299216 1,2 %A A299216 _R. H. Hardin_, Feb 05 2018 %I A299215 %S A299215 1,5,22,427,21730,3096833,1226106440,1213390065190,3070232362267950 %N A299215 Number of nXn 0..1 arrays with every element equal to 0, 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299215 Diagonal of A299221. %e A299215 Some solutions for n=5 %e A299215 ..0..1..1..0..1. .0..1..0..1..1. .0..0..0..1..0. .0..0..0..1..1 %e A299215 ..0..0..1..0..0. .1..1..1..1..0. .1..0..0..0..0. .0..1..0..1..1 %e A299215 ..1..0..0..0..1. .0..0..1..1..1. .0..0..0..1..0. .1..1..0..1..0 %e A299215 ..1..1..1..1..1. .1..0..1..1..1. .0..0..0..1..1. .0..0..1..0..0 %e A299215 ..0..1..0..1..0. .1..1..1..1..0. .0..1..0..1..1. .1..0..1..1..0 %Y A299215 Cf. A299221. %K A299215 nonn,new %O A299215 1,2 %A A299215 _R. H. Hardin_, Feb 05 2018 %I A298806 %S A298806 1,4,10,25,60,148,358,869,2106,5110,12396,30070,72942,176939,429214, %T A298806 1041172,2525640,6126607,14861710,36051016,87451296,212136296, %U A298806 514592810,1248281249,3028037016,7345306340,17817987338,43222250797,104847025002,254334247970,616955127612,1496588180810,3630371290710 %N A298806 Growth series for group with presentation < S, T : S^3 = T^6 = (S*T)^6 = 1 >. %H A298806 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298806 Index entries for linear recurrences with constant coefficients, signature (3,-2,1,2,-3,2,1,-2,3,-1). %F A298806 G.f.: (x^10 + x^9 + 2*x^7 - x^6 + 3*x^5 - x^4 + 2*x^3 + x + 1)/(x^10 - 3*x^9 + 2*x^8 - x^7 - 2*x^6 + 3*x^5 - 2*x^4 - x^3 + 2*x^2 - 3*x + 1). %F A298806 a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) + 2*a(n-4) - 3*a(n-5) + 2*a(n-6) + a(n-7) - 2*a(n-8) + 3*a(n-9) - a(n-10) for n>10. - _Colin Barker_, Feb 06 2018 %o A298806 (MAGMA) See Magma program in A298805. %o A298806 (PARI) Vec((1 - x + x^2)*(1 + x + x^2)*(1 + x - x^2 + x^3 - x^4 + x^5 + x^6) / (1 - 3*x + 2*x^2 - x^3 - 2*x^4 + 3*x^5 - 2*x^6 - x^7 + 2*x^8 - 3*x^9 + x^10) + O(x^40)) \\ _Colin Barker_, Feb 06 2018 %Y A298806 Cf. A008579, A298802, A298805. %K A298806 nonn,easy,new %O A298806 0,2 %A A298806 _John Cannon_ and _N. J. A. Sloane_, Feb 04 2018 %I A298805 %S A298805 1,3,4,6,8,12,16,22,24,34,40,56,62,83,98,133,152,202,236,322,368,496, %T A298805 570,776,892,1202,1384,1871,2158,2915,3352,4534,5218,7060,8120,10976, %U A298805 12636,17084,19664,26580,30592,41367,47604,64365,74072,100152,115264,155836,179352,242488,279076,377324,434246,587126 %N A298805 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^7 = 1 >. %H A298805 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298805 Index entries for linear recurrences with constant coefficients, signature (-1,0,0,1,2,1,0,1,0,1,2,1,0,0,-1,-1). %F A298805 G.f.: (-2*x^18 - 2*x^17 + 3*x^16 + 6*x^15 + 9*x^14 + 12*x^13 + 15*x^12 + 19*x^11 + 21*x^10 + 21*x^9 + 21*x^8 + 21*x^7 + 17*x^6 + 15*x^5 + 13*x^4 + 10*x^3 + 7*x^2 + 4*x + 1)/(x^16 + x^15 - x^12 - 2*x^11 - x^10 - x^8 - x^6 - 2*x^5 - x^4 + x + 1). %F A298805 The denominator can be factored: G.f. also = -(2*x^18+2*x^17-3*x^16-6*x^15-9*x^14-12*x^13-15*x^12-19*x^11-21*x^10-21*x^9-21*x^8-21*x^7-17*x^6-15*x^5-13*x^4-10*x^3-7*x^2-4*x-1)/((x^4+x^3+x^2+x+1)*(x^12-x^10-x^8+x^6-x^4-x^2+1)). %F A298805 a(n) = -a(n-1) + a(n-4) + 2*a(n-5) + a(n-6) + a(n-8) + a(n-10) + 2*a(n-11) + a(n-12) - a(n-15) - a(n-16) for n>18. - _Colin Barker_, Feb 06 2018 %o A298805 (MAGMA) %o A298805 // To get the growth function for the group with presentation %o A298805 // < S, T | S^a = T^b = (S*I)^c = 1 > %o A298805 a:=2; b:=3; c:=7; %o A298805 R := RationalFunctionField(Integers()); %o A298805 PSR := PowerSeriesRing(Integers():Precision := 100); %o A298805 FG := FreeGroup(2); %o A298805 TG := quo; %o A298805 f, A :=IsAutomaticGroup(TG); %o A298805 gf := GrowthFunction(A); %o A298805 R!gf; %o A298805 Coefficients(PSR!gf); %o A298805 (PARI) Vec((1 + 4*x + 7*x^2 + 10*x^3 + 13*x^4 + 15*x^5 + 17*x^6 + 21*x^7 + 21*x^8 + 21*x^9 + 21*x^10 + 19*x^11 + 15*x^12 + 12*x^13 + 9*x^14 + 6*x^15 + 3*x^16 - 2*x^17 - 2*x^18) / ((1 + x + x^2 + x^3 + x^4)*(1 - x^2 - x^4 + x^6 - x^8 - x^10 + x^12)) + O(x^60)) \\ _Colin Barker_, Feb 06 2018 %Y A298805 Cf. A008579, A298802. %K A298805 nonn,easy,new %O A298805 0,2 %A A298805 _John Cannon_ and _N. J. A. Sloane_, Feb 04 2018 %I A298879 %S A298879 0,3,6,12,13,15,21,24,26,27,30,37,42,45,47,48,52,53,54,57,60,61,63,69, %T A298879 73,74,81,83,84,90,93,94,96,99,104,105,106,107,108,109,111,114,115, %U A298879 119,120,122,123,126,133,137,138,141,146,148,151,155,159 %N A298879 Numbers whose square is not odious. %C A298879 Complement of A235331. %C A298879 From _Robert Israel_, Feb 02 2018: (Start) %C A298879 2*n is in the sequence if and only if n is in the sequence. %C A298879 2*n+1 is in the sequence if and only if n*(n+1) is odious. (End) %p A298879 select(t -> convert(convert(t^2,base,2),`+`)::even, [$0..200]); # _Robert Israel_, Feb 02 2018 %t A298879 Join[{0}, Select[Range[200], !OddQ[DigitCount[#^2, 2][[1]]] &]] %o A298879 (MAGMA) [n: n in [0..200] | IsEven(&+Intseq(n^2, 2))]; %o A298879 (PARI) isok(n) = (hammingweight(n^2) % 2) != 1; \\ _Michel Marcus_, Jan 31 2018 %Y A298879 Cf. A235001, A235331. %K A298879 nonn,easy,new %O A298879 1,2 %A A298879 _Vincenzo Librandi_, Jan 31 2018 %I A299114 %S A299114 3,4,5,6,8,10 %N A299114 Number of sides of a face of an Archimedean solid. %C A299114 Values of n for which the regular n-gon is a face of some Archimedean solid. %C A299114 Remarkably, the same is true for Johnson solids. Indeed, before Johnson (1966) and Zalgaller (1967) classified the 92 Johnson solids, Grünbaum and Johnson (1965) proved that the only polygons that occur as faces of a non-uniform regular-faced convex polyhedron (i.e., a Johnson solid) are triangles, squares, pentagons, hexagons, octagons, and decagons. %H A299114 Branko Grünbaum, Norman Johnson, The faces of a regular-faced polyhedron, J. Lond. Math. Soc. 40, 577-586 (1965). %H A299114 Norman W. Johnson, Convex Polyhedra with Regular Faces, Canadian Journal of Mathematics, 18 (1966), 169-200. %H A299114 Joseph Malkevitch, Regular-Faced Polyhedra: Remembering Norman Johnson, AMS Feature Column, Jan. 2018. %H A299114 Eric Weisstein's World of Mathematics, Archimedean Solid %H A299114 Wikipedia, List of Johnson solids %H A299114 Victor A. Zalgaller, Convex Polyhedra with Regular Faces, Zap. Nauchn. Sem. LOMI, 1967, Volume 2. Pages 5-221 (Mi znsl1408). %Y A299114 Cf. A092536, A092537, A092538, A242731, A242732, A242733. %K A299114 nonn,fini,full,new %O A299114 1,1 %A A299114 _Jonathan Sondow_, Feb 02 2018 %I A296169 %S A296169 1,1,6,59,810,14281,307566,7825859,229715130,7640988961,284037675966, %T A296169 11669182625099,525040651527210,25676859334384441,1356133254350401806, %U A296169 76928506160117877779,4664746297141400237850,301102611588796277314321,20613405033136513233790686,1491812049486032067219356699,113798761459974922574012320650 %N A296169 E.g.f. A(x) satisfies: A(x) = 1+x - cos(2*A(x) - x). %H A296169 Paul D. Hanna, Table of n, a(n) for n = 1..300 %F A296169 E.g.f. A(x) satisfies: %F A296169 (1) A(x) = 1+x - cos(2*A(x) - x). %F A296169 (2) A(x) = x + 2*sin(A(x) - x/2)^2. %F A296169 (3) A(x) = x/2 + Series_Reversion( 2*x + 2*cos(2*x) - 2 ). %F A296169 (4) A(x) = x/2 + Series_Reversion( 2*x - 4*sin(x)^2 ). %e A296169 E.g.f.: A(x) = x + x^2/2! + 6*x^3/3! + 59*x^4/4! + 810*x^5/5! + 14281*x^6/6! + 307566*x^7/7! + 7825859*x^8/8! + 229715130*x^9/9! + 7640988961*x^10/10! + ... %e A296169 such that A(x) = 1+x - cos(2*A(x) - x). %t A296169 terms = 21; A[_] = 0; Do[A[x_] = 1 + x - Cos[2*A[x] - x] + O[x]^(terms+1) // Normal, {terms+1}]; CoefficientList[A[x], x]*Range[0, terms]! // Rest (* _Jean-François Alcover_, Feb 05 2018 *) %o A296169 (PARI) {a(n) = my(A = x/2 + serreverse(2*x - 4*sin(x +x*O(x^n))^2) ); n!*polcoeff(A,n)} %o A296169 for(n=1,20,print1(a(n),", ")) %K A296169 nonn,new %O A296169 1,3 %A A296169 _Paul D. Hanna_, Feb 05 2018 %I A298232 %S A298232 1,3,17,41,10,6,77,33,7,8,28,167,1292,382,58,14,37,192,97,89,94,59,26, %T A298232 161,141,1187,71,22,148,3847,63,79,281,95,308,66,81,90,57,2387,288, %U A298232 1697,319,1786,669,30,173,1315,3626,924,20,447,67,2588,352,593,418,86,293,98 %N A298232 The decimal expansion of the fractional part of a(n)/a(n+1) starts with a(n+1) (disregarding leading zeros); always choose the smallest possible positive integer not occurring earlier. %C A298232 Numbers which can only appear as the first term of this sequence or the corresponding variant: 1, 2, 4, 5, 9, 11, 12, 13, 15, 16, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 34, 35, 38, 39, 43, 44, 45, 46, 47, 48, 49, etc. A298981 - _Robert G. Wilson v_, Jan 17 2018 %C A298232 The sequence is infinite. There will always be a solution of the form floor(sqrt(a(n)*10^k)) with k sufficiently large (namely, choose k such that this is larger than a(n) and the fractional part is < 0.5). - _M. F. Hasler_, Jan 17 2018 %C A298232 Records: 1, 3, 17, 41, 77, 167, 1292, 3847, 80498, 83666, 390256, 536097, 886566, 2533515, 4881598, 275680975, 7581556568, 10669182255, ... - _Robert G. Wilson v_, Jan 28 2018 %C A298232 If the restriction of a(n) not being a term previously present is removed then the sequence will cycle {3, 17, 41, 10}. - _Robert G. Wilson v_, Feb 04 2018 %H A298232 Jean-Marc Falcoz and Robert G. Wilson v, Table of n, a(n) for n = 1..6688(first 1001 terms from Jean-Marc Falcoz). %e A298232 1 divided by 3 is 0.3333333333... which shows "3" immediately after the decimal point; %e A298232 3 divided by 17 is 0.1764705882... which shows "17" immediately after the decimal point; %e A298232 17 divided by 41 is 0.4146341463... which shows "41" immediately after the decimal point; %e A298232 41 divided by 10 is 4.1000000000... which shows "10" immediately after the decimal point; %e A298232 10 divided by 6 is 1.6666666666... which shows "6" immediately after the decimal point; %e A298232 6 divided by 77 is 0.07792207792... which shows "77" after the decimal point and the leading zero; %e A298232 etc. %t A298232 f[s_List] := Block[{k = 2, m = s[[-1]]}, While[k = g[k, m]; MemberQ[s, k], k++]; Append[s, k]]; g[k_, m_] := Block[{j, l = k}, While[j = 10^IntegerLength[l]*Mod[m, l]/l; While[0 < Floor@j < l, j *= 10]; Floor[j] != l, l++]; l]; Nest[f, {1}, 100] (* _Robert G. Wilson v_, Jan 16 2018 and revised Jan 31 2018 *) %o A298232 (PARI) {u=[a=1]; (nxt()=for(b=u[1]+1,oo, !setsearch(u,b) && (f=frac(a/b)) && f\10^(-logint((b-1)\f,10)-1)==b&&return(b))); for(i=2,200, print1(a,","); u=setunion(u,[a=nxt()]));a} \\ _M. F. Hasler_, Jan 17 2018 %K A298232 nonn,base,new %O A298232 1,2 %A A298232 _Eric Angelini_ and _Jean-Marc Falcoz_, Jan 15 2018 [corrected by _Rémy Sigrist_ and _Jacques Tramu_, Jan 16 2018] %E A298232 a(2456) > 600000000. - _Robert G. Wilson v_, Jan 18 2018 %E A298232 a(2456) <= 7581556568. - _M. F. Hasler_, Jan 19 2018 %I A299168 %S A299168 1,0,0,0,5,6,42,64,387,5480,10461,113256,507390,1071084,4882635, %T A299168 44984560,382362589,891350154,7469477771,33066211100,78673599501, %U A299168 649785780710,2884039365010,22986956007816,306912836483025,1361558306986280,3519406658042964 %N A299168 Number of ordered ways of writing n-th prime number as a sum of n primes. %F A299168 a(n) = [x^prime(n)] (Sum_{k>=1} x^prime(k))^n. %e A299168 a(5) = 5 because fifth prime number is 11 and we have [3, 2, 2, 2, 2], [2, 3, 2, 2, 2], [2, 2, 3, 2, 2], [2, 2, 2, 3, 2] and [2, 2, 2, 2, 3]. %t A299168 Table[SeriesCoefficient[Sum[x^Prime[k], {k, 1, n}]^n, {x, 0, Prime[n]}], {n, 1, 27}] %Y A299168 Cf. A000040, A000586, A000607, A023360, A056768, A070215, A073610, A098238, A117278, A121303, A219180, A224344, A259254, A265112. %K A299168 nonn,new %O A299168 1,5 %A A299168 _Ilya Gutkovskiy_, Feb 04 2018 %I A299197 %S A299197 1,0,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0,2,1,0, %T A299197 0,0,1,1,1,1,0,0,0,2,2,0,0,0,1,1,2,1,0,0,0,2,2,1,0,0,0,2,2,2,0,0,1,1, %U A299197 3,1,1,0,0,2,3,2,0,0,1,2,3,2,1,0,0,2,3,3,0,0,0,1,4,3,1,0,0,2,3,3,2,0,0,1,4,4 %N A299197 Number of partitions of n into distinct parts that are greater of twin primes (A006512). %H A299197 Eric Weisstein's World of Mathematics, Twin Primes %H A299197 Index entries for related partition-counting sequences %F A299197 G.f.: Product_{k>=1} (1 + x^A006512(k)). %e A299197 a(31) = 2 because we have [31] and [19, 7, 5]. %t A299197 nmax = 105; CoefficientList[Series[Product[1 + Boole[PrimeQ[k] && PrimeQ[k - 2]] x^k, {k, 1, nmax}], {x, 0, nmax}], x] %Y A299197 Cf. A000586, A000607, A001097, A006512, A077608, A129363, A283875, A283876, A299196. %K A299197 nonn,new %O A299197 0,32 %A A299197 _Ilya Gutkovskiy_, Feb 04 2018 %I A299196 %S A299196 1,0,0,1,0,1,0,0,1,0,0,1,0,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,1,1,1, %T A299196 1,0,1,1,0,0,1,1,0,1,1,1,2,0,1,2,0,1,1,0,1,1,0,2,1,1,2,1,2,1,1,1,1,1, %U A299196 0,1,2,1,1,2,2,2,2,1,2,2,0,2,1,0,2,1,1,3,2,1,3,2,2,2,0,2,2,0,1,2,2,2,2,3,3,3 %N A299196 Number of partitions of n into distinct parts that are lesser of twin primes (A001359). %H A299196 Ilya Gutkovskiy, Scatter plot of a(n) - A299197(n) %H A299196 Eric Weisstein's World of Mathematics, Twin Primes %H A299196 Index entries for related partition-counting sequences %F A299196 G.f.: Product_{k>=1} (1 + x^A001359(k)). %e A299196 a(46) = 2 because we have [41, 5] and [29, 17]. %t A299196 nmax = 105; CoefficientList[Series[Product[1 + Boole[PrimeQ[k] && PrimeQ[k + 2]] x^k, {k, 1, nmax}], {x, 0, nmax}], x] %Y A299196 Cf. A000586, A000607, A001097, A001359, A077608, A129363, A283875, A283876, A299197. %K A299196 nonn,new %O A299196 0,47 %A A299196 _Ilya Gutkovskiy_, Feb 04 2018 %I A299195 %S A299195 1,1,0,0,0,30,6,0,0,0,360,157080,0,12586860,0,714233520,579379361, %T A299195 48062263014,46026944529624,759085890469938,170947379002578290, %U A299195 3331302954541376850,479526242126281889924,11322897238957194004884,1341983461418984670506352,31585668052999315295625900 %N A299195 Number of ordered ways of writing n^4 as a sum of n fourth powers of positive integers. %H A299195 Eric Weisstein's World of Mathematics, Biquadratic Number %F A299195 a(n) = [x^(n^4)] (Sum_{k>=1} x^(k^4))^n. %e A299195 a(6) = 6 because we have [256, 256, 256, 256, 256, 16], [256, 256, 256, 256, 16, 256], [256, 256, 256, 16, 256, 256], [256, 256, 16, 256, 256, 256], [256, 16, 256, 256, 256, 256] and [16, 256, 256, 256, 256, 256]. %t A299195 a[0] = 1; a[n_] := Coefficient[Sum[x^k^4, {k, n-1}]^n // Expand, x, n^4]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 25}] (* _Jean-François Alcover_, Feb 05 2018 *) %Y A299195 Cf. A000583, A046042, A259793, A298330, A298672, A298859, A299169. %K A299195 nonn,new %O A299195 0,6 %A A299195 _Ilya Gutkovskiy_, Feb 04 2018 %E A299195 More terms from _Alois P. Heinz_, Feb 04 2018 %I A297997 %S A297997 3,4,5,6,8,10,12,13,14,16,17,19,20,23,25,26,27,30,31,32,35,36,37,38, %T A297997 39,41,44,46,47,48,49,50,52,54,55,56,57,58,60,62,64,66,67,68,70,71,72, %U A297997 73,75,78,80,82,84,85,86,88,89,92,94,96,98,99,100,102,103 %N A297997 Solution (b(n)) of the near-complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments. %C A297997 The sequence (a(n)) generated by the equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + n, with initial values as shown, includes duplicates; e.g. a(18) = a(19) = 51. If the duplicates are removed from (a(n)), the resulting sequence and (b(n)) are complementary. Conjectures: %C A297997 (1) 1 <= b(k) - b(k-1) <= 3 for k>=1; %C A297997 (2) if d is in {1,2,3}, then b(k) = b(k-1) + d for infinitely many k. %C A297997 *** %C A297997 See A297830 for a guide to related sequences. %H A297997 Clark Kimberling, Table of n, a(n) for n = 0..9999 %e A297997 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 7. %e A297997 Complement: (b(n)) = (3, 4, 5, 6, 8,10,12,13,14,16, ...) %t A297997 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]); %t A297997 tbl = {}; a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; %t A297997 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + n; %t A297997 b[n_] := b[n] = mex[tbl = Join[{a[n], a[n - 1], b[n - 1]}, tbl], b[n - 1]]; %t A297997 Table[a[n], {n, 0, 300}] (* A297826 *) %t A297997 Table[b[n], {n, 0, 300}] (* A297997 *) %t A297997 (* _Peter J. C. Moses_, Jan 03 2017 *) %Y A297997 Cf. A297997, A297830. %K A297997 nonn,easy,new %O A297997 0,1 %A A297997 _Clark Kimberling_, Feb 04 2018 %I A297827 %S A297827 1,5,2,2,4,3,3,1,2,4,1,4,1,6,2,1,2,6,0,2,6,0,2,2,2,4,5,2,1,2,2,2,4,3, %T A297827 1,2,2,2,4,3,3,3,1,2,4,1,2,2,4,5,2,3,3,1,2,4,1,6,2,3,3,1,2,4,1,4,1,4, %U A297827 1,6,2,1,2,6,2,3,1,2,4,1,2,2,4,3,1,4 %N A297827 Difference sequence of A297826. %C A297827 Conjectures: %C A297827 (1) 0 <= a(k) <= 6 for k>=1; %C A297827 (2) if d is in {0,1,2,3,4,5,6}, then a(k) = d for infinitely many k; for d = 0, see A297829. %H A297827 Clark Kimberling, Table of n, a(n) for n = 1..10000 %t A297827 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]); %t A297827 tbl = {}; a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; %t A297827 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + n; %t A297827 b[n_] := b[n] = mex[tbl = Join[{a[n], a[n - 1], b[n - 1]}, tbl], b[n - 1]]; %t A297827 u = Table[a[n], {n, 0, 300}](* A297826 *) %t A297827 v = Table[b[n], {n, 0, 300}](* A297997 *) %t A297827 Differences[u]; (* A297827 *) %t A297827 Differences[v]; (* A297828 *) %t A297827 (* _Peter J. C. Moses_, Jan 03 2017 *) %Y A297827 Cf. A297826, A297828. %K A297827 nonn,easy,new %O A297827 1,2 %A A297827 _Clark Kimberling_, Feb 04 2018 %I A297826 %S A297826 1,2,7,9,11,15,18,21,22,24,28,29,33,34,40,42,43,45,51,51,53,59,59,61, %T A297826 63,65,69,74,76,77,79,81,83,87,90,91,93,95,97,101,104,107,110,111,113, %U A297826 117,118,120,122,126,131,133,136,139,140,142,146,147,153,155 %N A297826 Solution (a(n)) of the near-complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments. %C A297826 The sequence (a(n)) generated by the equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + n, with initial values as shown, includes duplicates; e.g. a(18) = a(19) = 51. If the duplicates are removed from (a(n)), the resulting sequence and (b(n)) are complementary. Conjectures: %C A297826 (1) 0 <= a(k) - a(k-1) <= 6 for k>=1; %C A297826 (2) if d is in {0,1,2,3,4,5,6}, then a(k) = a(k-1) + d for infinitely many k. %C A297826 *** %C A297826 See A297830 for a guide to related sequences. %H A297826 Clark Kimberling, Table of n, a(n) for n = 0..10000 %e A297826 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 7. %e A297826 Complement: (b(n)) = (3, 4, 5, 6, 8,10,12,13,14,16, ...) %t A297826 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]); %t A297826 tbl = {}; a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; %t A297826 a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + n; %t A297826 b[n_] := b[n] = mex[tbl = Join[{a[n], a[n - 1], b[n - 1]}, tbl], b[n - 1]]; %t A297826 Table[a[n], {n, 0, 300}] (* A297826 *) %t A297826 Table[b[n], {n, 0, 300}] (* A297997 *) %t A297826 (* _Peter J. C. Moses_, Jan 03 2017 *) %Y A297826 Cf. A297997, A297830. %K A297826 nonn,easy,new %O A297826 0,2 %A A297826 _Clark Kimberling_, Feb 04 2018 %I A299020 %S A299020 1,1,2,2,4,1,6,3,3,2,10,2,12,3,3,3,16,4,18,1,4,6,22,1,7,9,5,5,28,4,30, %T A299020 4,7,9,4,3,36,13,8,3,40,5,42,8,4,15,46,3,11,6,12,9,52,6,8,6,15,15,58, %U A299020 2,60,22,5,6,7,9,66,12,17,4,70,4,72,31,5,14,7 %N A299020 a(n) is the maximum digit in the factorial base expansion of 1/n. %C A299020 See the Wikipedia link for the construction method of 1/n in factorial base. %H A299020 Rémy Sigrist, Table of n, a(n) for n = 1..10000 %H A299020 Rémy Sigrist, Colored scatterplot of the first 25000 terms (where the color is function of A052126(n)) %H A299020 Wikipedia, Factorial number system (Fractional values) %H A299020 Index entries for sequences related to factorial base representation %F A299020 a(n!) = 1 for any n >= 0. %F A299020 a(n! / k) = k for any n > 1 and k = 1..n-1. %F A299020 a(p) = p - 1 for any prime p. %e A299020 The first terms, alongside 1/n in factorial base, are: %e A299020 n a(n) 1/n in factorial base %e A299020 -- ---- --------------------- %e A299020 1 1 1 %e A299020 2 1 0.1 %e A299020 3 2 0.0 2 %e A299020 4 2 0.0 1 2 %e A299020 5 4 0.0 1 0 4 %e A299020 6 1 0.0 1 %e A299020 7 6 0.0 0 3 2 0 6 %e A299020 8 3 0.0 0 3 %e A299020 9 3 0.0 0 2 3 2 %e A299020 10 2 0.0 0 2 2 %e A299020 11 10 0.0 0 2 0 5 3 1 4 0 10 %e A299020 12 2 0.0 0 2 %e A299020 13 12 0.0 0 1 4 1 2 5 4 8 5 0 12 %e A299020 14 3 0.0 0 1 3 3 3 %e A299020 15 3 0.0 0 1 3 %e A299020 16 3 0.0 0 1 2 3 %e A299020 17 16 0.0 0 1 2 0 2 3 6 8 9 0 9 2 7 0 16 %e A299020 18 4 0.0 0 1 1 4 %e A299020 19 18 0.0 0 1 1 1 6 2 0 9 5 2 6 11 11 13 8 0 18 %e A299020 20 1 0.0 0 1 1 %t A299020 a[n_] := Module[{m = 0, r = 1, f = 1/n}, While[f > 0, m = Max[m, Floor[f]]; r++; f = FractionalPart[f]*r]; m]; Array[a, 77] (* _Jean-François Alcover_, Feb 05 2018, translated from PARI *) %o A299020 (PARI) a(n) = my (m=0, r=1, f=1/n); while (f>0, m = max(m, floor(f)); r++; f = frac(f)*r); return (m) %Y A299020 Cf. A052126, A246359, A276350. %K A299020 nonn,base,new %O A299020 1,3 %A A299020 _Rémy Sigrist_, Jan 31 2018 %I A299025 %S A299025 0,5,52,2,521,1,5260,50,40,52130,520,20,526510,5210,10,800,5218700, %T A299025 52600,500,400,52609300,521300,5200,200,521359100,6100,5265100,52100, %U A299025 100,5265679000,8000,52187000,526000,5000,52182884000,4000,526093000,23000,5213000,52000 %N A299025 a(n) = the fractional part of 1 / A003592(n) read backwards. %C A299025 Numbers in this sequence that also appear in A003592, sorted, include the product of numbers k | 10^e with integer e >= 0 and 10^m with m >= e. For instance, the proper divisors of 10 {1, 2, 5} appear and {10, 20, 40, 50} follow, finally {100, 200, 400, 500, 800} followed by any product k 10^m with k = {1, 2, 4, 5, 8} and m >= 3. - _Michael De Vlieger_, Feb 03 2018 %H A299025 Rémy Sigrist, Table of n, a(n) for n = 1..10000 %F A299025 a(A180953(n)) = 10^(n-1) for any n > 0. %e A299025 The first terms, alongside A003592(n) and the fractional part of 1/A003592(n), are: %e A299025 n a(n) A003592(n) frac(1/A003592(n)) %e A299025 -- ---- ---------- ------------------ %e A299025 1 0 1 0 %e A299025 2 5 2 0.5 %e A299025 3 52 4 0.25 %e A299025 4 2 5 0.2 %e A299025 5 521 8 0.125 %e A299025 6 1 10 0.1 %e A299025 7 5260 16 0.0625 %e A299025 8 50 20 0.05 %e A299025 9 40 25 0.04 %e A299025 10 52130 32 0.03125 %e A299025 11 520 40 0.025 %e A299025 12 20 50 0.02 %e A299025 13 526510 64 0.015625 %e A299025 14 5210 80 0.0125 %e A299025 15 10 100 0.01 %e A299025 16 800 125 0.008 %e A299025 17 5218700 128 0.0078125 %e A299025 18 52600 160 0.00625 %e A299025 19 500 200 0.005 %e A299025 20 400 250 0.004 %t A299025 With[{e = 12}, Table[FromDigits@ Reverse@ PadLeft[#1, Length@ #1 + Abs@ #2] - 10 Boole[n == 1] & @@ RealDigits[1/n], {n, Sort@ Flatten@ Table[2^i*5^j, {i, 0, e}, {j, 0, Log[5, 2^(e - i)]}]}]] (* _Michael De Vlieger_, Feb 03 2018, after _Robert G. Wilson v_ at A003592 *) %o A299025 (PARI) mx = 4000; A003592 = vecsort(concat(vector(1+logint(mx,2), i, vector(1+logint(floor(mx/2^(i-1)), 5), j, 2^(i-1) * 5^(j-1))))) %o A299025 backward(n) = my (v=0, i=frac(1/n), r=1/10); while (i, v += r*floor(i); i=frac(i)*10; r*=10); v %o A299025 print (apply(backward, A003592)) %Y A299025 Cf. A003592, A004086, A180953. %K A299025 nonn,base,easy,new %O A299025 1,2 %A A299025 _Rémy Sigrist_, Feb 01 2018 %I A298696 %S A298696 1,1,3,28,410,8386,220962,7140736,273712896,12146997564,612813677300, %T A298696 34647736132384,2170381958609592,149223874286440552, %U A298696 11173356309069883320,905099760309260722560,78870011549256151244288,7356892186010414244194704,731435433368215011644979504,77216368897429504869064200256,8626428901029156775683110378400,1016792561657783042048699052986016 %N A298696 G.f.: Sum_{n>=0} binomial(n*(n+1), n)/(n+1) * x^n / (1 + x)^(n*(n+1)). %C A298696 Compare g.f. to: 1 = Sum_{n>=0} binomial(m*(n+1), n)/(n+1) * x^n / (1+x)^(m*(n+1)) holds for fixed m. %H A298696 Paul D. Hanna, Table of n, a(n) for n = 0..300 %F A298696 a(n) ~ c * d^n * n! / n^2, where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.176554609483480358231680164050876553672889... and c = 0.226094037474708064867716267720651240574569526310006844420310030408773601638... - _Vaclav Kotesovec_, Feb 07 2018 %e A298696 G.f.: A(x) = 1 + x + 3*x^2 + 28*x^3 + 410*x^4 + 8386*x^5 + 220962*x^6 + 7140736*x^7 + 273712896*x^8 + 12146997564*x^9 + 612813677300*x^10 + ... %e A298696 such that %e A298696 A(x) = 1 + C(2,1)/2*x/(1+x)^2 + C(6,2)/3*x^2/(1+x)^6 + C(12,3)/4*x^3/(1+x)^12 + C(20,4)/5*x^4/(1+x)^20 + C(30,5)/6*x^5/(1+x)^30 + ... %e A298696 more explicitly, %e A298696 A(x) = 1 + x/(1+x)^2 + 5*x^2/(1+x)^6 + 55*x^3/(1+x)^12 + 969*x^4/(1+x)^20 + 23751*x^5/(1+x)^30 + ... + A135861(n)*x^n/(1+x)^(n*(n+1)) + ... %t A298696 terms = 22; s = Sum[Binomial[n*(n + 1), n]/(n + 1)*x^n/(1 + x)^(n*(n + 1)), {n, 0, terms}] + O[x]^terms; CoefficientList[s, x] (* _Jean-François Alcover_, Feb 05 2018 *) %o A298696 (PARI) {a(n) = my(A = sum(m=0,n,binomial(m*(m+1),m)/(m+1)*x^m/(1+x +x*O(x^n))^(m*(m+1)) ) ); polcoeff(A,n)} %o A298696 for(n=0,25, print1(a(n),", ")) %Y A298696 Cf. A298695, A135861. %K A298696 nonn,new %O A298696 0,3 %A A298696 _Paul D. Hanna_, Feb 04 2018 %I A298695 %S A298695 1,1,5,61,1123,27671,853411,31603447,1365807689,67469763889, %T A298695 3749935785301,231591200859701,15733654527061483,1166102347943957815, %U A298695 93629607937879486019,8096167402408961507311,750088483178476669111441,74127049788588758257392161,7783440821906363883725443813,865349148215025766722403077229,101553078711812924877087765912371 %N A298695 G.f.: Sum_{n>=0} binomial(n^2, n) * x^n / (1 + x)^(n^2). %C A298695 Compare g.f. to: Sum_{n>=0} binomial(m*n, n) * x^n / (1+x)^(m*n) = (1+x)/(1 - (m-1)*x) holds for fixed m. %H A298695 Paul D. Hanna, Table of n, a(n) for n = 0..300 %e A298695 G.f.: A(x) = 1 + x + 5*x^2 + 61*x^3 + 1123*x^4 + 27671*x^5 + 853411*x^6 + 31603447*x^7 + 1365807689*x^8 + 67469763889*x^9 + 3749935785301*x^10 + ... %e A298695 such that %e A298695 A(x) = 1 + C(1,1)*x/(1+x) + C(4,2)*x^2/(1+x)^4 + C(9,3)*x^3/(1+x)^9 + C(16,4)*x^4/(1+x)^16 + C(25,5)*x^5/(1+x)^25 + C(36,6)*x^6/(1+x)^36 + ... %e A298695 more explicitly, %e A298695 A(x) = 1 + x/(1+x) + 6*x^2/(1+x)^4 + 84*x^3/(1+x)^9 + 1820*x^4/(1+x)^16 + 53130*x^5/(1+x)^25 + 1947792*x^6/(1+x)^36 + ... + A014062(n)*x^n/(1+x)^(n^2) + ... %t A298695 terms = 21; s = Sum[Binomial[n^2, n]*x^n/(1 + x)^(n^2), {n, 0, terms}] + O[x]^terms; CoefficientList[s, x] (* _Jean-François Alcover_, Feb 06 2018 *) %o A298695 (PARI) {a(n) = my(A = sum(m=0,n,binomial(m^2,m)*x^m/(1+x +x*O(x^n))^(m^2) ) ); polcoeff(A,n)} %o A298695 for(n=0,25, print1(a(n),", ")) %Y A298695 Cf. A298696, A014062. %K A298695 nonn,new %O A298695 0,3 %A A298695 _Paul D. Hanna_, Feb 04 2018 %I A299194 %S A299194 1,2,2,3,4,3,5,3,3,5,8,13,3,13,8,13,34,9,9,34,13,21,73,19,80,19,73,21, %T A299194 34,203,59,220,220,59,203,34,55,594,129,518,1539,518,129,594,55,89, %U A299194 1443,355,2466,3704,3704,2466,355,1443,89,144,4013,891,8609,25097,11459,25097 %N A299194 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299194 Table starts %C A299194 ..1....2...3.....5......8......13.......21........34.........55...........89 %C A299194 ..2....4...3....13.....34......73......203.......594.......1443.........4013 %C A299194 ..3....3...3.....9.....19......59......129.......355........891.........2317 %C A299194 ..5...13...9....80....220.....518.....2466......8609......26954.......108253 %C A299194 ..8...34..19...220...1539....3704....25097....161188.....638942......3784977 %C A299194 .13...73..59...518...3704...11459....80188....533274....2432710.....15899938 %C A299194 .21..203.129..2466..25097...80188...848789...8080772...44495267....401953734 %C A299194 .34..594.355..8609.161188..533274..8080772.122310309..832353461..10703053961 %C A299194 .55.1443.891.26954.638942.2432710.44495267.832353461.6602925250.104043829095 %H A299194 R. H. Hardin, Table of n, a(n) for n = 1..199 %F A299194 Empirical for column k: %F A299194 k=1: a(n) = a(n-1) +a(n-2) %F A299194 k=2: a(n) = a(n-1) +3*a(n-2) +8*a(n-3) -4*a(n-4) -16*a(n-5) for n>6 %F A299194 k=3: [order 17] for n>18 %F A299194 k=4: [order 69] for n>70 %e A299194 Some solutions for n=5 k=4 %e A299194 ..0..0..1..1. .0..1..1..0. .0..1..1..0. .0..1..1..1. .0..0..1..0 %e A299194 ..0..0..0..0. .0..1..1..1. .1..0..1..1. .1..0..1..1. .1..1..1..1 %e A299194 ..1..1..1..1. .1..1..1..0. .0..0..1..0. .0..0..0..0. .1..1..1..1 %e A299194 ..0..0..1..1. .0..1..1..1. .0..0..1..1. .0..0..0..0. .0..0..1..0 %e A299194 ..1..1..1..0. .0..1..1..0. .1..1..1..0. .1..0..0..1. .0..0..0..1 %Y A299194 Column 1 is A000045(n+1). %Y A299194 Column 2 is A297901. %K A299194 nonn,tabl,new %O A299194 1,2 %A A299194 _R. H. Hardin_, Feb 04 2018 %I A299193 %S A299193 21,203,129,2466,25097,80188,848789,8080772,44495267,401953734, %T A299193 3543442090,25212181714,216188808140,1850221884503,14678677380641, %U A299193 124620964817279,1059123538097555,8784723556861615,74638466920690089 %N A299193 Number of nX7 0..1 arrays with every element equal to 0, 1, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299193 Column 7 of A299194. %H A299193 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299193 Some solutions for n=5 %e A299193 ..0..0..0..0..0..1..1. .0..1..1..0..0..1..1. .0..1..0..1..0..0..1 %e A299193 ..0..0..1..1..0..0..0. .0..1..1..0..0..0..0. .0..1..0..1..0..0..1 %e A299193 ..1..1..1..1..1..1..1. .0..0..0..0..0..0..0. .0..0..0..0..0..0..0 %e A299193 ..1..1..1..0..1..0..0. .0..0..0..0..0..1..1. .0..0..0..0..0..0..0 %e A299193 ..0..0..1..1..1..1..1. .1..0..1..1..0..0..0. .1..0..0..1..1..0..1 %Y A299193 Cf. A299194. %K A299193 nonn,new %O A299193 1,1 %A A299193 _R. H. Hardin_, Feb 04 2018 %I A299192 %S A299192 13,73,59,518,3704,11459,80188,533274,2432710,15899938,99781878, %T A299192 547440363,3435785232,21267682619,125513673832,779215203891, %U A299192 4812218593453,29261363183192,181267978405834,1121025211720547 %N A299192 Number of nX6 0..1 arrays with every element equal to 0, 1, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299192 Column 6 of A299194. %H A299192 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299192 Some solutions for n=5 %e A299192 ..0..0..1..1..1..0. .0..0..0..0..0..0. .0..0..0..1..1..1. .0..1..0..0..1..1 %e A299192 ..0..0..1..1..0..1. .0..0..1..0..1..1. .0..0..1..0..1..1. .1..0..0..0..0..0 %e A299192 ..1..1..1..1..1..1. .1..1..1..1..1..1. .1..1..1..0..0..0. .0..0..0..0..0..0 %e A299192 ..1..1..1..1..1..0. .0..0..0..0..0..0. .0..0..0..1..1..1. .0..1..0..1..1..0 %e A299192 ..0..0..1..1..0..1. .1..1..0..0..1..1. .0..0..1..0..1..1. .0..1..0..1..1..0 %Y A299192 Cf. A299194. %K A299192 nonn,new %O A299192 1,1 %A A299192 _R. H. Hardin_, Feb 04 2018 %I A299191 %S A299191 8,34,19,220,1539,3704,25097,161188,638942,3784977,21950458,107591478, %T A299191 603715833,3379272687,17836462387,98596630709,545047497960, %U A299191 2953569300978,16258588851100,89536091522140,489737577922388,2693288014288399 %N A299191 Number of nX5 0..1 arrays with every element equal to 0, 1, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299191 Column 5 of A299194. %H A299191 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299191 Some solutions for n=5 %e A299191 ..0..0..0..0..1. .0..1..1..0..1. .0..1..1..0..0. .0..1..0..1..1 %e A299191 ..1..1..0..1..0. .1..0..1..1..0. .0..1..1..1..1. .1..0..0..0..0 %e A299191 ..0..0..0..0..0. .0..0..0..0..0. .0..0..1..1..1. .0..0..0..0..0 %e A299191 ..0..0..0..0..0. .1..0..1..1..0. .1..0..1..1..0. .1..0..0..1..1 %e A299191 ..1..1..0..0..1. .0..1..1..1..0. .0..1..1..1..0. .1..0..0..1..1 %Y A299191 Cf. A299194. %K A299191 nonn,new %O A299191 1,1 %A A299191 _R. H. Hardin_, Feb 04 2018 %I A299190 %S A299190 5,13,9,80,220,518,2466,8609,26954,108253,391026,1364326,5142635, %T A299190 18805585,67927770,251558219,923421565,3375978025,12442380432, %U A299190 45744080554,167955473490,618311475706,2274757925559,8365332143494 %N A299190 Number of nX4 0..1 arrays with every element equal to 0, 1, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299190 Column 4 of A299194. %H A299190 R. H. Hardin, Table of n, a(n) for n = 1..210 %H A299190 R. H. Hardin, Empirical recurrence of order 69 %F A299190 Empirical recurrence of order 69 (see link above) %e A299190 Some solutions for n=5 %e A299190 ..0..1..0..0. .0..0..1..1. .0..0..1..1. .0..0..0..0. .0..1..1..0 %e A299190 ..1..1..1..1. .0..0..0..0. .1..1..1..1. .0..0..1..1. .0..1..1..1 %e A299190 ..1..1..1..1. .1..1..1..1. .1..1..1..0. .1..1..1..1. .1..1..1..0 %e A299190 ..0..1..1..0. .1..1..0..0. .0..0..1..1. .1..1..1..0. .0..1..1..1 %e A299190 ..0..1..1..0. .0..0..0..0. .1..1..1..1. .0..1..1..0. .0..1..1..0 %Y A299190 Cf. A299194. %K A299190 nonn,new %O A299190 1,1 %A A299190 _R. H. Hardin_, Feb 04 2018 %I A299189 %S A299189 3,3,3,9,19,59,129,355,891,2317,6019,15543,40557,105639,274475,716909, %T A299189 1868955,4870915,12717105,33177547,86555071,225940757,589641087, %U A299189 1538803595,4016626149,10483511171,27362265959,71420814253,186418208451 %N A299189 Number of nX3 0..1 arrays with every element equal to 0, 1, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299189 Column 3 of A299194. %H A299189 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299189 Empirical: a(n) = a(n-1) +4*a(n-2) +13*a(n-3) -13*a(n-4) -46*a(n-5) -59*a(n-6) +53*a(n-7) +153*a(n-8) +98*a(n-9) -60*a(n-10) -140*a(n-11) -7*a(n-12) +67*a(n-13) +15*a(n-14) -78*a(n-15) -48*a(n-16) -32*a(n-17) for n>18 %e A299189 Some solutions for n=5 %e A299189 ..0..1..1. .0..1..1. .0..1..0. .0..0..1. .0..0..1. .0..1..1. .0..1..0 %e A299189 ..1..1..1. .0..1..1. .0..1..0. .0..0..1. .0..0..0. .0..1..1. .0..1..0 %e A299189 ..1..0..1. .0..0..0. .0..0..0. .0..1..1. .0..1..0. .0..0..1. .1..1..1 %e A299189 ..1..1..1. .0..1..1. .0..0..0. .0..0..1. .0..0..0. .0..1..1. .0..1..0 %e A299189 ..0..1..1. .0..1..1. .1..0..1. .0..0..1. .1..0..0. .0..1..1. .0..1..0 %Y A299189 Cf. A299194. %K A299189 nonn,new %O A299189 1,1 %A A299189 _R. H. Hardin_, Feb 04 2018 %I A299188 %S A299188 1,4,3,80,1539,11459,848789,122310309,6602925250,2215354712154 %N A299188 Number of nXn 0..1 arrays with every element equal to 0, 1, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299188 Diagonal of A299194. %e A299188 Some solutions for n=5 %e A299188 ..0..0..1..0..1. .0..0..1..1..0. .0..1..0..0..1. .0..0..1..0..0 %e A299188 ..0..0..0..0..0. .1..1..1..1..1. .1..0..0..1..0. .0..0..1..0..0 %e A299188 ..1..0..0..0..0. .1..1..1..1..1. .1..1..1..1..1. .1..1..0..1..1 %e A299188 ..0..0..0..1..1. .0..0..1..0..0. .1..1..1..1..1. .0..0..1..0..0 %e A299188 ..1..0..0..0..0. .1..1..1..1..1. .0..1..1..0..0. .0..0..1..0..0 %Y A299188 Cf. A299194. %K A299188 nonn,new %O A299188 1,2 %A A299188 _R. H. Hardin_, Feb 04 2018 %I A299187 %S A299187 1,2,2,4,7,4,8,13,13,8,16,29,20,29,16,32,69,27,27,69,32,64,137,47,75, %T A299187 47,137,64,128,301,83,191,191,83,301,128,256,705,137,401,626,401,137, %U A299187 705,256,512,1461,235,952,1442,1442,952,235,1461,512,1024,3193,412,2258,4115 %N A299187 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299187 Table starts %C A299187 ...1....2...4....8....16.....32......64.....128......256.......512.......1024 %C A299187 ...2....7..13...29....69....137.....301.....705.....1461......3193.......7373 %C A299187 ...4...13..20...27....47.....83.....137.....235......412.......709.......1228 %C A299187 ...8...29..27...75...191....401.....952....2258.....5275.....13250......32268 %C A299187 ..16...69..47..191...626...1442....4115...12839....34387....101533.....306593 %C A299187 ..32..137..83..401..1442...3773...12847...48408...157805....562522....2057541 %C A299187 ..64..301.137..952..4115..12847...59287..270914..1088697...5017429...22861814 %C A299187 .128..705.235.2258.12839..48408..270914.1666826..8168186..46505377..277299195 %C A299187 .256.1461.412.5275.34387.157805.1088697.8168186.51863841.370258161.2763281188 %H A299187 R. H. Hardin, Table of n, a(n) for n = 1..220 %F A299187 Empirical for column k: %F A299187 k=1: a(n) = 2*a(n-1) %F A299187 k=2: a(n) = 3*a(n-1) -2*a(n-2) +8*a(n-3) -20*a(n-4) +8*a(n-5) for n>6 %F A299187 k=3: [order 17] for n>19 %F A299187 k=4: [order 67] for n>70 %e A299187 Some solutions for n=5 k=4 %e A299187 ..0..1..0..1. .0..0..0..0. .0..0..1..0. .0..1..0..0. .0..0..1..1 %e A299187 ..0..0..0..1. .1..0..0..1. .1..1..0..1. .0..1..1..1. .1..0..1..0 %e A299187 ..0..0..1..0. .0..1..0..0. .0..0..0..1. .1..0..1..0. .1..1..1..0 %e A299187 ..1..0..0..1. .1..0..0..0. .1..0..1..0. .1..0..1..0. .1..1..1..0 %e A299187 ..1..1..0..1. .1..0..1..0. .1..0..1..0. .1..0..1..0. .1..0..0..1 %Y A299187 Column 1 is A000079(n-1). %Y A299187 Column 2 is A297883. %K A299187 nonn,tabl,new %O A299187 1,2 %A A299187 _R. H. Hardin_, Feb 04 2018 %I A299186 %S A299186 64,301,137,952,4115,12847,59287,270914,1088697,5017429,22861814, %T A299186 97586048,444232955,2021643018,8825115683,39848981071,180691585774, %U A299186 798045866431,3587277159543,16208542978994,72063515690216 %N A299186 Number of nX7 0..1 arrays with every element equal to 0, 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299186 Column 7 of A299187. %H A299186 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299186 Some solutions for n=5 %e A299186 ..0..1..1..1..1..0..0. .0..0..0..1..0..0..0. .0..1..0..0..1..0..1 %e A299186 ..0..1..0..1..1..1..1. .0..1..0..1..0..1..0. .1..0..1..1..0..1..0 %e A299186 ..0..1..1..1..1..0..0. .0..0..0..1..0..0..0. .1..0..0..0..0..1..1 %e A299186 ..0..1..1..1..1..0..1. .0..0..0..1..0..0..0. .0..0..1..1..0..0..0 %e A299186 ..0..1..0..0..1..0..0. .1..0..0..0..0..0..1. .1..1..0..0..1..1..0 %Y A299186 Cf. A299187. %K A299186 nonn,new %O A299186 1,1 %A A299186 _R. H. Hardin_, Feb 04 2018 %I A299185 %S A299185 32,137,83,401,1442,3773,12847,48408,157805,562522,2057541,7039840, %T A299185 25174801,91158763,318102256,1134299958,4083010670,14386161345, %U A299185 51212037306,183601965542,650165124338,2313003043768,8270512715806 %N A299185 Number of nX6 0..1 arrays with every element equal to 0, 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299185 Column 6 of A299187. %H A299185 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299185 Some solutions for n=5 %e A299185 ..0..1..0..0..1..1. .0..1..0..1..1..1. .0..1..1..1..0..1. .0..1..0..0..1..0 %e A299185 ..1..0..0..0..0..0. .1..1..0..0..0..0. .1..1..1..0..0..1. .0..0..0..0..0..1 %e A299185 ..0..0..0..1..0..1. .1..1..0..0..0..1. .1..1..1..0..0..1. .0..0..1..0..0..0 %e A299185 ..0..0..0..1..1..0. .0..1..0..1..1..1. .0..1..1..1..0..1. .1..0..0..0..0..0 %e A299185 ..0..1..1..0..1..0. .0..0..0..1..0..0. .1..1..1..0..1..0. .1..0..1..0..0..1 %Y A299185 Cf. A299187. %K A299185 nonn,new %O A299185 1,1 %A A299185 _R. H. Hardin_, Feb 04 2018 %I A299184 %S A299184 16,69,47,191,626,1442,4115,12839,34387,101533,306593,857981,2513530, %T A299184 7495901,21435358,62395272,184826545,533845224,1551901224,4572367457, %U A299184 13275628433,38607493908,113319776343,329919021938,960178593422 %N A299184 Number of nX5 0..1 arrays with every element equal to 0, 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299184 Column 5 of A299187. %H A299184 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299184 Some solutions for n=5 %e A299184 ..0..0..1..1..1. .0..1..0..0..0. .0..1..0..0..1. .0..1..0..1..0 %e A299184 ..0..1..1..1..0. .1..0..0..0..1. .0..0..1..1..0. .0..1..1..1..0 %e A299184 ..0..0..0..0..0. .0..0..0..0..1. .1..1..0..1..1. .1..0..1..1..0 %e A299184 ..1..1..0..0..0. .0..0..1..0..1. .1..0..1..1..1. .0..1..1..1..1 %e A299184 ..1..0..0..0..1. .0..1..0..1..0. .1..1..1..1..0. .1..0..0..0..0 %Y A299184 Cf. A299187. %K A299184 nonn,new %O A299184 1,1 %A A299184 _R. H. Hardin_, Feb 04 2018 %I A299183 %S A299183 8,29,27,75,191,401,952,2258,5275,13250,32268,77769,191931,469537, %T A299183 1146722,2829540,6941097,16987427,41832684,102763482,252007870, %U A299183 619897823,1523184474,3738568449,9191254923,22587057264,55465702765,136320314139 %N A299183 Number of nX4 0..1 arrays with every element equal to 0, 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299183 Column 4 of A299187. %H A299183 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299183 Empirical: a(n) = 3*a(n-1) +a(n-2) +11*a(n-3) -49*a(n-4) -18*a(n-5) +11*a(n-6) +218*a(n-7) +80*a(n-8) -244*a(n-9) +84*a(n-10) -286*a(n-11) -825*a(n-12) -1687*a(n-13) +2587*a(n-14) +4682*a(n-15) -576*a(n-16) -6980*a(n-17) +1611*a(n-18) +10469*a(n-19) -3111*a(n-20) -19296*a(n-21) -3761*a(n-22) +16046*a(n-23) +2091*a(n-24) -2785*a(n-25) +718*a(n-26) +21722*a(n-27) -16663*a(n-28) +13289*a(n-29) -14849*a(n-30) -23694*a(n-31) -14617*a(n-32) +65721*a(n-33) -25438*a(n-34) -46244*a(n-35) +41970*a(n-36) +102361*a(n-37) -104134*a(n-38) -38629*a(n-39) +29336*a(n-40) +56521*a(n-41) -83376*a(n-42) -20392*a(n-43) +63523*a(n-44) -2523*a(n-45) +4025*a(n-46) -9755*a(n-47) +11795*a(n-48) -3658*a(n-49) -4777*a(n-50) +5165*a(n-51) -9718*a(n-52) +4047*a(n-53) +2228*a(n-54) -1986*a(n-55) +2919*a(n-56) -1872*a(n-57) -1708*a(n-58) +367*a(n-59) -180*a(n-60) +69*a(n-61) +272*a(n-62) +76*a(n-63) +32*a(n-64) +14*a(n-65) -24*a(n-66) -8*a(n-67) for n>70 %e A299183 Some solutions for n=5 %e A299183 ..0..0..1..1. .0..0..1..0. .0..1..0..1. .0..1..0..1. .0..0..1..0 %e A299183 ..1..0..1..0. .1..0..1..0. .0..0..0..1. .0..1..1..0. .1..0..1..0 %e A299183 ..1..1..1..0. .1..1..1..1. .0..0..0..1. .1..0..1..0. .1..1..1..1 %e A299183 ..1..1..1..0. .1..1..1..1. .0..1..0..1. .0..1..1..0. .1..1..1..1 %e A299183 ..0..1..1..1. .1..0..0..1. .0..0..0..1. .0..1..0..1. .0..1..1..0 %Y A299183 Cf. A299187. %K A299183 nonn,new %O A299183 1,1 %A A299183 _R. H. Hardin_, Feb 04 2018 %I A299182 %S A299182 4,13,20,27,47,83,137,235,412,709,1228,2150,3758,6578,11556,20330, %T A299182 35805,63163,111574,197307,349294,618999,1098034,1949511,3464203, %U A299182 6160917,10965304,19530386,34810655,62087577,110808092,197881858,353585148 %N A299182 Number of nX3 0..1 arrays with every element equal to 0, 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299182 Column 3 of A299187. %H A299182 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299182 Empirical: a(n) = 3*a(n-1) -a(n-2) +3*a(n-3) -14*a(n-4) +a(n-5) +4*a(n-6) +23*a(n-7) +3*a(n-8) -10*a(n-9) -21*a(n-10) -4*a(n-11) +3*a(n-13) +a(n-14) +6*a(n-15) +6*a(n-16) +4*a(n-17) for n>19 %e A299182 Some solutions for n=5 %e A299182 ..0..1..0. .0..1..1. .0..1..0. .0..1..1. .0..0..0. .0..0..0. .0..1..0 %e A299182 ..0..0..0. .0..1..0. .0..0..0. .0..0..0. .0..1..0. .0..1..0. .1..0..0 %e A299182 ..1..1..1. .0..1..1. .0..0..0. .1..1..0. .0..0..0. .0..0..0. .0..0..0 %e A299182 ..1..0..1. .0..1..1. .0..1..0. .0..0..0. .1..1..1. .1..1..1. .0..0..1 %e A299182 ..1..1..1. .0..1..0. .0..0..0. .0..1..1. .1..0..1. .0..0..0. .0..1..0 %Y A299182 Cf. A299187. %K A299182 nonn,new %O A299182 1,1 %A A299182 _R. H. Hardin_, Feb 04 2018 %I A299181 %S A299181 1,7,20,75,626,3773,59287,1666826,51863841,3412264531 %N A299181 Number of nXn 0..1 arrays with every element equal to 0, 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299181 Diagonal of A299187. %e A299181 Some solutions for n=5 %e A299181 ..0..1..1..0..1. .0..1..0..0..1. .0..0..1..0..0. .0..1..0..0..0 %e A299181 ..1..0..0..1..1. .0..0..0..0..0. .1..0..1..1..0. .1..0..0..0..1 %e A299181 ..0..0..1..0..0. .0..0..0..0..0. .0..1..0..1..1. .0..0..0..0..0 %e A299181 ..0..0..0..0..1. .0..1..0..1..0. .1..0..1..1..1. .1..1..1..0..0 %e A299181 ..1..0..0..1..1. .1..1..0..1..1. .0..0..1..0..1. .0..0..0..1..0 %Y A299181 Cf. A299187. %K A299181 nonn,new %O A299181 1,2 %A A299181 _R. H. Hardin_, Feb 04 2018 %I A299180 %S A299180 1,2,2,4,8,4,8,25,25,8,16,85,70,85,16,32,286,205,205,286,32,64,969, %T A299180 614,649,614,969,64,128,3281,1860,2153,2153,1860,3281,128,256,11114, %U A299180 5631,7016,8368,7016,5631,11114,256,512,37649,17034,22819,30089,30089,22819 %N A299180 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299180 Table starts %C A299180 ...1.....2.....4......8......16......32.......64.......128........256 %C A299180 ...2.....8....25.....85.....286.....969.....3281.....11114......37649 %C A299180 ...4....25....70....205.....614....1860.....5631.....17034......51507 %C A299180 ...8....85...205....649....2153....7016....22819.....73931.....239461 %C A299180 ..16...286...614...2153....8368...30089...105709....376826....1341575 %C A299180 ..32...969..1860...7016...30089..123258...488812...1973932....7998917 %C A299180 ..64..3281..5631..22819..105709..488812..2207509..10286880...48492179 %C A299180 .128.11114.17034..73931..376826.1973932.10286880..56481101..314537796 %C A299180 .256.37649.51507.239461.1341575.7998917.48492179.314537796.2090883077 %H A299180 R. H. Hardin, Table of n, a(n) for n = 1..264 %F A299180 Empirical for column k: %F A299180 k=1: a(n) = 2*a(n-1) %F A299180 k=2: a(n) = 3*a(n-1) +a(n-2) +2*a(n-3) -2*a(n-4) -4*a(n-5) %F A299180 k=3: [order 11] for n>12 %F A299180 k=4: [order 22] for n>28 %e A299180 Some solutions for n=5 k=5 %e A299180 ..0..0..1..0..1. .0..1..1..0..1. .0..0..1..0..1. .0..1..1..0..0 %e A299180 ..1..0..0..1..0. .0..1..1..0..1. .1..1..1..1..0. .0..1..1..0..1 %e A299180 ..0..0..0..0..1. .1..1..1..1..1. .1..1..1..1..1. .1..1..1..1..1 %e A299180 ..1..0..0..0..0. .0..1..1..1..0. .0..0..1..1..0. .0..1..1..1..0 %e A299180 ..0..1..0..1..0. .1..1..0..1..1. .0..1..0..1..1. .1..1..0..1..1 %Y A299180 Column 1 is A000079(n-1). %Y A299180 Column 2 is A281338. %Y A299180 Column 3 is A298282. %K A299180 nonn,tabl,new %O A299180 1,2 %A A299180 _R. H. Hardin_, Feb 04 2018 %I A299179 %S A299179 64,3281,5631,22819,105709,488812,2207509,10286880,48492179,225721140, %T A299179 1052511651,4943137693,23205542625,108821827316,511190051629, %U A299179 2403886138964,11305007633490,53192247798072,250461934195726 %N A299179 Number of nX7 0..1 arrays with every element equal to 0, 1, 2, 3, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299179 Column 7 of A299180. %H A299179 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299179 Some solutions for n=5 %e A299179 ..0..1..0..1..1..0..0. .0..0..0..0..1..1..0. .0..1..0..1..1..0..0 %e A299179 ..1..0..1..0..1..0..1. .1..1..1..1..0..0..1. .1..0..0..0..0..0..1 %e A299179 ..1..0..1..0..1..0..1. .1..0..0..1..0..1..1. .0..0..0..0..0..0..0 %e A299179 ..1..0..1..0..1..1..0. .1..0..0..1..0..1..0. .1..0..0..1..0..0..1 %e A299179 ..1..0..0..1..0..0..0. .1..1..1..1..0..1..0. .0..1..0..0..1..0..0 %Y A299179 Cf. A299180. %K A299179 nonn,new %O A299179 1,1 %A A299179 _R. H. Hardin_, Feb 04 2018 %I A299178 %S A299178 32,969,1860,7016,30089,123258,488812,1973932,7998917,32133289, %T A299178 129295516,521825491,2103866573,8477867045,34181235225,137822897266, %U A299178 555658803759,2240383895918,9033674194617,36426046739915,146881842890670 %N A299178 Number of nX6 0..1 arrays with every element equal to 0, 1, 2, 3, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299178 Column 6 of A299180. %H A299178 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299178 Some solutions for n=5 %e A299178 ..0..1..0..1..1..1. .0..1..0..1..1..1. .0..1..0..1..0..0. .0..0..1..1..0..1 %e A299178 ..1..0..0..0..0..1. .1..0..1..1..0..0. .1..0..1..1..1..1. .1..0..1..1..0..1 %e A299178 ..0..0..0..0..0..1. .1..1..1..1..1..1. .1..1..1..1..1..1. .1..1..1..1..1..0 %e A299178 ..1..0..0..1..1..1. .0..1..1..1..1..1. .1..0..1..1..0..0. .1..0..1..1..1..1 %e A299178 ..0..1..0..0..0..0. .1..1..0..1..0..0. .1..0..1..1..0..0. .0..1..0..1..0..1 %Y A299178 Cf. A299180. %K A299178 nonn,new %O A299178 1,1 %A A299178 _R. H. Hardin_, Feb 04 2018 %I A299177 %S A299177 16,286,614,2153,8368,30089,105709,376826,1341575,4761555,16938266, %T A299177 60287898,214388007,762687683,2714155126,9656982501,34362409576, %U A299177 122287872312,435186845623,1548741989760,5511952960843,19617235873777 %N A299177 Number of nX5 0..1 arrays with every element equal to 0, 1, 2, 3, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299177 Column 5 of A299180. %H A299177 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299177 Some solutions for n=5 %e A299177 ..0..0..1..0..1. .0..0..1..0..1. .0..0..1..0..1. .0..0..1..0..0 %e A299177 ..1..1..1..1..0. .1..0..0..1..0. .1..1..1..1..0. .1..0..0..0..1 %e A299177 ..1..1..1..1..1. .0..0..0..0..0. .1..1..1..1..1. .0..0..0..0..0 %e A299177 ..0..0..1..1..0. .1..0..0..0..1. .0..0..1..1..0. .1..0..0..1..0 %e A299177 ..1..0..1..1..0. .0..1..0..1..0. .0..0..1..1..0. .0..0..1..0..1 %Y A299177 Cf. A299180. %K A299177 nonn,new %O A299177 1,1 %A A299177 _R. H. Hardin_, Feb 04 2018 %I A299176 %S A299176 8,85,205,649,2153,7016,22819,73931,239461,777197,2523034,8188618, %T A299176 26573796,86235511,279858388,908244102,2947587609,9565980797, %U A299176 31045039874,100752470139,326979037002,1061168684227,3443888186471,11176703761069 %N A299176 Number of nX4 0..1 arrays with every element equal to 0, 1, 2, 3, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299176 Column 4 of A299180. %H A299176 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299176 Empirical: a(n) = 6*a(n-1) -10*a(n-2) +6*a(n-3) -15*a(n-4) +31*a(n-5) -44*a(n-6) +33*a(n-7) +22*a(n-8) +97*a(n-9) -84*a(n-10) -139*a(n-11) -109*a(n-12) +111*a(n-13) +183*a(n-14) -32*a(n-15) +12*a(n-16) -113*a(n-17) +47*a(n-18) +33*a(n-19) -40*a(n-20) +26*a(n-21) -20*a(n-22) for n>28 %e A299176 Some solutions for n=6 %e A299176 ..0..1..0..0. .0..1..0..0. .0..0..1..0. .0..1..1..1. .0..0..0..0 %e A299176 ..0..0..0..1. .0..0..0..1. .1..0..0..0. .0..1..0..0. .1..1..0..1 %e A299176 ..1..0..0..0. .1..0..0..0. .0..0..0..1. .0..0..0..1. .0..1..1..1 %e A299176 ..0..0..0..1. .0..0..0..1. .1..0..0..0. .1..0..0..0. .1..1..1..0 %e A299176 ..0..1..0..0. .1..0..0..0. .0..0..0..1. .0..0..0..1. .0..1..1..1 %e A299176 ..0..1..1..1. .0..0..1..0. .0..1..0..0. .0..1..0..0. .1..1..0..1 %Y A299176 Cf. A299180. %K A299176 nonn,new %O A299176 1,1 %A A299176 _R. H. Hardin_, Feb 04 2018 %I A299175 %S A299175 1,8,70,649,8368,123258,2207509,56481101,2090883077,106695657776, %T A299175 7917434629718 %N A299175 Number of nXn 0..1 arrays with every element equal to 0, 1, 2, 3, 6 or 7 king-move adjacent elements, with upper left element zero. %C A299175 Diagonal of A299180. %e A299175 Some solutions for n=5 %e A299175 ..0..1..1..0..1. .0..1..0..1..0. .0..1..0..1..0. .0..0..1..0..1 %e A299175 ..1..0..1..1..1. .1..0..0..0..1. .0..0..0..0..1. .1..1..1..1..0 %e A299175 ..1..1..1..1..0. .0..0..0..0..0. .1..0..0..0..0. .1..1..1..1..1 %e A299175 ..0..1..1..1..1. .0..1..0..0..1. .0..0..0..1..0. .0..0..1..1..0 %e A299175 ..1..0..1..0..1. .1..1..0..0..1. .0..1..0..0..1. .0..1..0..1..1 %Y A299175 Cf. A299180. %K A299175 nonn,new %O A299175 1,2 %A A299175 _R. H. Hardin_, Feb 04 2018 %I A296338 %S A296338 1,0,0,1,1,0,0,0,1,0,0,0,1,1,0,1,0,0,0,0,0,0,0,0,2,0,0,0,1,1,0,0,0,0, %T A296338 0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,1,1,0,0,0,0,0,1,0,0,1,0,0,0,0, %U A296338 0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,1,0,0,0,1,1 %N A296338 a(n) = number of partitions of n into consecutive positive squares. %F A296338 a(A034705(n)) >= 1 for n > 1. %e A296338 1 = 1^2, so a(1) = 1. %e A296338 4 = 2^2, so a(4) = 1. %e A296338 5 = 1^2 + 2^2, so a(5) = 1. %e A296338 9 = 3^2, so a(9) = 1. %e A296338 13 = 2^2 + 3^2, so a(13) = 1. %e A296338 14 = 1^2 + 2^2 + 3^2, so a(14) = 1. %e A296338 16 = 4^2, so a(16) = 1. %e A296338 25 = 3^2 + 4^2 = 5^2, so a(25) = 2. %e A296338 29 = 2^2 + 3^2 + 4^2, so a(29) = 1. %e A296338 30 = 1^2 + 2^2 + 3^2 + 4^2, so a(30) = 1. %t A296338 nMax = 100; t = {0}; Do[k = n; s = 0; While[s = s + k^2; s <= nMax, AppendTo[t, s]; k++], {n, 1, nMax}]; tt = Tally[t]; a[_] = 0; Do[a[tt[[i, 1]]] = tt[[i, 2]], {i, 1, Length[tt]}]; Table[a[n], {n, 1, nMax}] (* _Jean-François Alcover_, Feb 04 2018, using _T. D. Noe_'s program for A034705 *) %o A296338 (Ruby) %o A296338 def A296338(n) %o A296338 m = Math.sqrt(n).to_i %o A296338 ary = Array.new(n + 1, 0) %o A296338 (1..m).each{|i| %o A296338 sum = i * i %o A296338 ary[sum] += 1 %o A296338 i += 1 %o A296338 sum += i * i %o A296338 while sum <= n %o A296338 ary[sum] += 1 %o A296338 i += 1 %o A296338 sum += i * i %o A296338 end %o A296338 } %o A296338 ary[1..-1] %o A296338 end %o A296338 p A296338(100) %Y A296338 Cf. A000290, A001227, A034705, A130052, A234304, A297199, A298467. %K A296338 nonn,new %O A296338 1,25 %A A296338 _Seiichi Manyama_, Jan 14 2018 %I A298948 %S A298948 1,0,-2,-2,1,2,1,0,2,2,-2,-6,-2,2,2,0,3,2,-1,-6,-2,2,3,-2,4,6,0,-10, %T A298948 -4,0,4,-2,5,8,6,-12,-6,-4,-1,-6,12,10,8,-12,-4,-4,1,-18,11,18,15,-20, %U A298948 -2,-8,7,-18,8,12,29,-24,2,-8,3,-34,21,6,29,-32,5,-8,31,-52 %N A298948 Expansion of Product_{k>=1} (1 - x^prime(k))^2. %C A298948 Self-convolution of A046675. %Y A298948 Cf. A046675, A280245, A298436. %K A298948 sign,new %O A298948 0,3 %A A298948 _Seiichi Manyama_, Jan 30 2018 %I A299165 %S A299165 1,0,0,8,0,0,0,24,27,20,12,24,112,126,120,48,180,324,140,0,420,460,24, %T A299165 360,275,510,0,532,720,390,672,96,318,406,840,612,68,630,144,1480, %U A299165 1260,1680,676,156,0,344,1296,1344,343,1800,72,392,1566,540,2520,1680 %N A299165 a(n) = A000594(n) mod n*(n+1). %H A299165 Seiichi Manyama, Table of n, a(n) for n = 1..10000 %o A299165 (PARI) {a(n) = ramanujantau(n)%(n*(n+1))} %Y A299165 Cf. A000594, A299158, A299163. %K A299165 nonn,new %O A299165 1,4 %A A299165 _Seiichi Manyama_, Feb 04 2018 %I A299163 %S A299163 1,0,0,3,0,0,0,6,7,9,0,11,0,6,8,14,0,1,0,0,2,0,0,10,15,24,0,10,0,18,0, %T A299163 30,12,21,12,20,30,6,24,4,0,3,16,21,0,15,0,21,43,15,20,21,0,45,0,27, %U A299163 42,34,0,28,46,42,56,38,48,60,16,0,14,63,0,50,60,36,12 %N A299163 a(n) = A000594(n) mod (n+1). %H A299163 Seiichi Manyama, Table of n, a(n) for n = 1..10000 %o A299163 (PARI) {a(n) = ramanujantau(n)%(n+1)} %Y A299163 Cf. A000594, A273650, A299157. %K A299163 nonn,new %O A299163 1,4 %A A299163 _Seiichi Manyama_, Feb 04 2018 %I A299158 %S A299158 2,3,5,6,7,20,27,45,91,160,240,243,343,384,792,896,2639,1163799 %N A299158 Numbers n such that n*(n+1) divides tau(n), where tau(n)=A000594(n) is Ramanujan's tau function. %C A299158 a(19) > 5*10^6. %C A299158 Numbers n such that A299165(n) = 0 %C A299158 Intersection of A063938 and A299157. %H A299158 Eric Weisstein's World of Mathematics, Tau Function %Y A299158 Cf. A000594, A063938, A299157, A299165. %K A299158 nonn,more,new %O A299158 1,1 %A A299158 _Seiichi Manyama_, Feb 04 2018 %I A299157 %S A299157 2,3,5,6,7,11,13,17,19,20,22,23,27,29,31,41,45,47,53,55,59,68,71,76, %T A299157 77,79,83,87,89,91,97,104,107,114,127,137,139,149,160,167,171,177,179, %U A299157 183,191,195,199,209,223,229,239,240,243,251,269,275,293,297,321,343 %N A299157 Numbers n such that n+1 divides tau(n), where tau(n)=A000594(n) is Ramanujan's tau function. %C A299157 Numbers n such that A299163(n) = 0. %H A299157 Seiichi Manyama, Table of n, a(n) for n = 1..500 %H A299157 Eric Weisstein's World of Mathematics, Tau Function %o A299157 (PARI) isok(n) = (ramanujantau(n) % (n+1)) == 0; \\ _Michel Marcus_, Feb 05 2018 %Y A299157 Cf. A000594, A063938, A079334, A296991, A299158, A299163. %K A299157 nonn,new %O A299157 1,1 %A A299157 _Seiichi Manyama_, Feb 04 2018 %I A297778 %S A297778 1,1,1,1,1,1,1,1,1,2,1,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,1,2, %T A297778 2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,1,2,2, %U A297778 2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2 %N A297778 Number of distinct runs in base-10 digits of n. %C A297778 Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. A043562(n) = a(n) for n=1..100, but not for n=101. %H A297778 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297778 14! in base-10: 2,7,0,0,1,7,3,0,1,3,0,0; ten runs, of which 6 are distinct, so that a(14!) = 6. %t A297778 b = 10; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297778 Table[s[n], {n, 1, 200}] %Y A297778 Cf. A043562 (number of runs, not necessarily distinct), A297770. %K A297778 nonn,base,easy,new %O A297778 1,10 %A A297778 _Clark Kimberling_, Feb 03 2018 %I A297784 %S A297784 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1, %T A297784 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1, %U A297784 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2 %N A297784 Number of distinct runs in base-16 digits of n. %C A297784 Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. %H A297784 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297784 17830144 in base-16: 1,1,0,1,1,0,0; four runs, of which 3 are distinct, so that a(17830144) = 3. %t A297784 b = 16; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297784 Table[s[n], {n, 1, 200}] %Y A297784 Cf. A043568 (number of runs, not necessarily distinct), A297770. %K A297784 nonn,base,easy,new %O A297784 1,16 %A A297784 _Clark Kimberling_, Feb 03 2018 %I A297783 %S A297783 1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2, %T A297783 2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2, %U A297783 2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2 %N A297783 Number of distinct runs in base-15 digits of n. %C A297783 Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. %H A297783 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297783 12153600 in base-15: 1,1,0,1,1,0,0; four runs, of which 3 are distinct, so that a(12153600) = 3. %t A297783 b = 15; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297783 Table[s[n], {n, 1, 200}] %Y A297783 Cf. A043567 (number of runs, not necessarily distinct), A297770. %K A297783 nonn,base,easy,new %O A297783 1,15 %A A297783 _Clark Kimberling_, Feb 03 2018 %I A297782 %S A297782 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2, %T A297782 2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2, %U A297782 2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2 %N A297782 Number of distinct runs in base-14 digits of n. %C A297782 Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. %H A297782 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297782 8070300 in base-14: 1,1,0,1,1,0,0; four runs, of which 3 are distinct, so that a(8070300) = 3. %t A297782 b = 14; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297782 Table[s[n], {n, 1, 200}] %Y A297782 Cf. A043566 (number of runs, not necessarily distinct), A297770. %K A297782 nonn,base,easy,new %O A297782 1,14 %A A297782 _Clark Kimberling_, Feb 03 2018 %I A297781 %S A297781 1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2, %T A297781 2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2, %U A297781 2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2 %N A297781 Number of distinct runs in base-13 digits of n. %C A297781 Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. %H A297781 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297781 5200468 in base-13: 1,1,0,1,1,0,0; four runs, of which 3 are distinct, so that a(5200468) = 3. %t A297781 b = 13; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297781 Table[s[n], {n, 1, 200}] %Y A297781 Cf. A043565 (number of runs, not necessarily distinct), A297770. %K A297781 nonn,base,easy,new %O A297781 1,13 %A A297781 _Clark Kimberling_, Feb 03 2018 %I A297780 %S A297780 1,1,1,1,1,1,1,1,1,1,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2, %T A297780 2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2, %U A297780 2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2 %N A297780 Number of distinct runs in base-12 digits of n. %C A297780 Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. %H A297780 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297780 3006865 in base-12: 1,0,1,0,1,0,1; seven runs, of which 2 are distinct, so that a(3006865) = 2. %t A297780 b = 12; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297780 Table[s[n], {n, 1, 200}] %Y A297780 Cf. A043564 (number of runs, not necessarily distinct), A297770. %K A297780 nonn,base,easy,new %O A297780 1,12 %A A297780 _Clark Kimberling_, Feb 03 2018 %I A297779 %S A297779 1,1,1,1,1,1,1,1,1,1,2,1,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2, %T A297779 2,1,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2, %U A297779 2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,1,2,2 %N A297779 Number of distinct runs in base-11 digits of n. %C A297779 Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. %H A297779 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297779 11! in base-11: 2,0,5,9,4,0,10,0; eight runs, of which 6 are distinct, so that a(11!) = 6. %t A297779 b = 11; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297779 Table[s[n], {n, 1, 200}] %Y A297779 Cf. A043563 (number of runs, not necessarily distinct), A297770. %K A297779 nonn,base,easy,new %O A297779 1,11 %A A297779 _Clark Kimberling_, Feb 03 2018 %I A299169 %S A299169 1,1,2,3,4,35,12,217,8,58473,7930,572891,5556,122985733,5175184, %T A299169 22299917655,579379377,743262257063,56837361641571,1395217574459461, %U A299169 375984668290604635 %N A299169 Number of ordered ways of writing n^4 as a sum of n fourth powers of nonnegative integers. %H A299169 Eric Weisstein's World of Mathematics, Biquadratic Number %F A299169 a(n) = [x^(n^4)] (Sum_{k>=0} x^(k^4))^n. %e A299169 a(6) = 12 because we have [1296, 0, 0, 0, 0, 0], [256, 256, 256, 256, 256, 16], [256, 256, 256, 256, 16, 256], [256, 256, 256, 16, 256, 256], [256, 256, 16, 256, 256, 256], [256, 16, 256, 256, 256, 256], [16, 256, 256, 256, 256, 256], [0, 1296, 0, 0, 0, 0], [0, 0, 1296, 0, 0, 0], [0, 0, 0, 1296, 0, 0], [0, 0, 0, 0, 1296, 0] and [0, 0, 0, 0, 0, 1296]. %Y A299169 Cf. A000583, A046042, A259793, A298329, A298671, A298859, A298989. %K A299169 nonn,more,new %O A299169 0,3 %A A299169 _Ilya Gutkovskiy_, Feb 04 2018 %I A299167 %S A299167 1,1,2,5,14,36,94,243,628,1619,4178,10776,27793,71682,184879,476832, %T A299167 1229830,3171942,8180989,21100215,54421187,140361900,362018270, %U A299167 933709453,2408202606,6211182512,16019743522,41317765457,106565859669,274852289679,708892898170,1828360759013,4715667307920 %N A299167 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)^k). %H A299167 N. J. A. Sloane, Transforms %F A299167 G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)^k). %F A299167 a(0) = 1; a(n) = Sum_{k=1..n} A026007(k-1)*a(n-k). %t A299167 nmax = 32; CoefficientList[Series[1/(1 - x Product[(1 + x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x] %Y A299167 Antidiagonal sums of A277938. %Y A299167 Cf. A026007, A067687, A299105, A299106, A299108, A299162, A299164, A299166. %K A299167 nonn,new %O A299167 0,3 %A A299167 _Ilya Gutkovskiy_, Feb 04 2018 %I A299166 %S A299166 1,1,2,6,17,48,132,365,1003,2759,7583,20843,57283,157442,432719, %T A299166 1189317,3268818,8984318,24693343,67869557,186539251,512702559, %U A299166 1409161449,3873076007,10645137706,29258128633,80415877302,221022792843,607480469466,1669658209311,4589050472041 %N A299166 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 - x^k)^k). %H A299166 Alois P. Heinz, Table of n, a(n) for n = 0..1000 %H A299166 N. J. A. Sloane, Transforms %F A299166 G.f.: 1/(1 - x*Product_{k>=1} 1/(1 - x^k)^k). %F A299166 a(0) = 1; a(n) = Sum_{k=1..n} A000219(k-1)*a(n-k). %p A299166 b:= proc(n, k) option remember; `if`(n=0, 1, k*add( %p A299166 b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n) %p A299166 end: %p A299166 a:= n-> add(b(n-j, j), j=0..n): %p A299166 seq(a(n), n=0..35); # _Alois P. Heinz_, Feb 04 2018 %t A299166 nmax = 30; CoefficientList[Series[1/(1 - x Product[1/(1 - x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x] %Y A299166 Antidiagonal sums of A255961. %Y A299166 Cf. A000219, A067687, A299105, A299106, A299108, A299162, A299164, A299167. %K A299166 nonn,new %O A299166 0,3 %A A299166 _Ilya Gutkovskiy_, Feb 04 2018 %I A299164 %S A299164 1,1,2,5,14,35,91,233,597,1517,3885,9922,25333,64683,165181,421828, %T A299164 1077277,2750993,7025168,17940298,45814165,116996152,298774246, %U A299164 762982615,1948434235,4975732669,12706571546,32448880807,82864981016,211613009498,540397935771,1380018797044,3524165721799 %N A299164 Expansion of 1/(1 - x*Product_{k>=1} (1 + k*x^k)). %H A299164 N. J. A. Sloane, Transforms %F A299164 G.f.: 1/(1 - x*Product_{k>=1} (1 + k*x^k)). %F A299164 a(0) = 1; a(n) = Sum_{k=1..n} A022629(k-1)*a(n-k). %t A299164 nmax = 32; CoefficientList[Series[1/(1 - x Product[1 + k x^k, {k, 1, nmax}]), {x, 0, nmax}], x] %Y A299164 Antidiagonal sums of A297321. %Y A299164 Cf. A022629, A067687, A299105, A299106, A299108, A299162, A299166, A299167. %K A299164 nonn,new %O A299164 0,3 %A A299164 _Ilya Gutkovskiy_, Feb 04 2018 %I A299162 %S A299162 1,1,2,6,17,49,135,380,1051,2925,8119,22548,62574,173767,482360, %T A299162 1339126,3717700,10321163,28653557,79548612,220843925,613110573, %U A299162 1702128034,4725475979,13118945083,36421037100,101112695940,280710759278,779313926949,2163544401343,6006468273440 %N A299162 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 - k*x^k)). %H A299162 N. J. A. Sloane, Transforms %F A299162 G.f.: 1/(1 - x*Product_{k>=1} 1/(1 - k*x^k)). %F A299162 a(0) = 1; a(n) = Sum_{k=1..n} A006906(k-1)*a(n-k). %t A299162 nmax = 30; CoefficientList[Series[1/(1 - x Product[1/(1 - k x^k), {k, 1, nmax}]), {x, 0, nmax}], x] %Y A299162 Antidiagonal sums of A297328. %Y A299162 Cf. A006906, A067687, A299105, A299106, A299108, A299164, A299166, A299167. %K A299162 nonn,new %O A299162 0,3 %A A299162 _Ilya Gutkovskiy_, Feb 04 2018 %I A299173 %S A299173 1,0,0,1,2,0,0,0,1,0,0,0,2,3,0,1,0,0,0,0,0,0,0,0,2,0,0,0,3,4,0,0,0,0, %T A299173 0,1,0,0,0,0,2,0,0,0,0,0,0,0,1,3,0,0,0,4,5,0,0,0,0,0,2,0,0,1,0,0,0,0, %U A299173 0,0,0,0,0,0,0,0,3,0,0,0,1,0,0,0,2,4,0,0,0,5,6,0,0,0,0,0,0,0,0,1 %N A299173 a(n) is the maximum number of squared consecutive positive integers into which the integer n can be partitioned. %C A299173 a(k^2)>=1, the inequality being strict if k is in A097812. %H A299173 Robert Israel, Table of n, a(n) for n = 1..10000 %e A299173 25 = 5^2 = 3^2 + 4^2 and no such partition is longer, so a(25) = 2. %e A299173 30 = 1^2 + 2^2 + 3^2 + 4^2 and no such partition is longer, so a(30) = 4. %e A299173 2018 = 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2 + 13^2 + 14^2 + 15^2 + 16^2 + 17^2 + 18^2 and no such partition is longer, so a(2018) = 12. (This special example is due to _Seiichi Manyama_.) - _Jean-François Alcover_, Feb 05 2018 %p A299173 N:= 200: # to get a(1)..a(N) %p A299173 A:= Vector(N): %p A299173 S:= n -> n*(n+1)*(2*n+1)/6: %p A299173 M:= floor(sqrt(N)): %p A299173 for d from 1 to M do %p A299173 for b from d to M do %p A299173 s:= S(b) - S(b-d); %p A299173 if s > N then break fi; %p A299173 A[s]:= d %p A299173 od od: %p A299173 convert(A,list); # _Robert Israel_, Feb 04 2018 %t A299173 terms = 100; jmax = Ceiling[Sqrt[terms]]; kmax = Ceiling[(3*terms)^(1/3)]; Clear[a]; a[_] = 0; Do[r = Range[j, j + k - 1]; n = r . r; If[k > a[n], a[n] = k], {j, jmax}, {k, kmax}]; Array[a, terms] %Y A299173 Cf. A034705, A097812, A130052, A111044, A234304, A234311, A296338, A298467. %K A299173 nonn,look,new %O A299173 1,5 %A A299173 _Jean-François Alcover_, Feb 04 2018 %I A298803 %S A298803 1,4,8,16,30,50,88,150,260,448,768,1328,2284,3930,6776,11662,20082, %T A298803 34592,59560,102570,176642,304180,523830,902084,1553452,2675184, %U A298803 4606892,7933444,13662066,23527220,40515838,69771678,120152672,206912968,356321478,613615442 %N A298803 Growth series for group with presentation < S, T : S^3 = T^3 = (S*T)^4 = 1 >. %H A298803 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298803 Index entries for linear recurrences with constant coefficients, signature (0,1,3,1,0,-1). %F A298803 G.f.: (1 + 4*x + 7*x^2 + 9*x^3 + 9*x^4 + 6*x^5 + 3*x^6 - 2*x^7) / ((1 + x + x^2)*(1 - x - x^2 - x^3 + x^4)). [Corrected by _Colin Barker_, Feb 04 2018] %F A298803 a(n) = a(n-2) + 3*a(n-3) + a(n-4) - a(n-6) for n>7. - _Colin Barker_, Feb 04 2018 %o A298803 (MAGMA) %o A298803 R := RationalFunctionField(Integers()); %o A298803 PSR25 := PowerSeriesRing(Integers():Precision := 25); %o A298803 FG := FreeGroup(2); %o A298803 TG := quo; %o A298803 f, A :=IsAutomaticGroup(TG); %o A298803 gf := GrowthFunction(A); %o A298803 R!gf; %o A298803 Coefficients(PSR25!gf); %o A298803 (PARI) Vec((1 + 4*x + 7*x^2 + 9*x^3 + 9*x^4 + 6*x^5 + 3*x^6 - 2*x^7) / ((1 + x + x^2)*(1 - x - x^2 - x^3 + x^4)) + O(x^40)) \\ _Colin Barker_, Feb 04 2018 %Y A298803 Cf. A008579, A298802. %K A298803 nonn,easy,new %O A298803 0,2 %A A298803 _John Cannon_ and _N. J. A. Sloane_, Feb 04 2018 %I A299082 %S A299082 1,5,6,7,4,6,8,2,5,5,7,7,4,0,5,3,0,7,4,8,6,3,3,4,0,3,8,4,1,7,9,6,8,8, %T A299082 4,4,4,5,0,2,9,3,3,0,5,8,5,4,8,8,2,8,3,4,0,2,9,7,9,6,5,3,1,9,1,2,1,5, %U A299082 6,2,3,6,8,4,8,6,7,2,2,8,5,9,6,5,0,2,9,2 %N A299082 Decimal expansion of Gamma(e). %F A299082 Gamma(e) = Integral_{x >= 0} x^(e-1)/e^x dx. %e A299082 1.56746825577405307486334038417968844450293305854882834029796531912156236... %p A299082 evalf(GAMMA(exp(1)),120); %t A299082 First@ RealDigits[Gamma@ E, 10, 105] (* _Michael De Vlieger_, Feb 03 2018 *) %o A299082 (PARI) gamma(exp(1)) \\ _Michel Marcus_, Feb 04 2018 %Y A299082 Cf. A001113, A202412, A269545. %K A299082 nonn,cons,new %O A299082 1,2 %A A299082 _Paolo P. Lava_, Feb 02 2018 %I A298802 %S A298802 1,4,10,24,56,128,294,676,1552,3564,8186,18800,43176,99160,227734, %T A298802 523020,1201184,2758676,6335658,14550664,33417496,76747632,176260934, %U A298802 404806196,929690160,2135154556,4903660570,11261895264,25864409480,59400985544,136422101046,313311125788,719559813184 %N A298802 Growth series for group with presentation < S, T : S^4 = T^4 = (S*T)^4 = 1 >. %H A298802 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298802 Index entries for linear recurrences with constant coefficients, signature (2,0,2,-1). %F A298802 G.f.: (1 + x)^2*(1 + x^2) / (1 - 2*x - 2*x^3 + x^4). %F A298802 a(n) = 2*a(n-1) + 2*a(n-3) - a(n-4) for n>4. - _Colin Barker_, Feb 04 2018 %o A298802 (MAGMA) %o A298802 R := RationalFunctionField(Integers()); %o A298802 PSR25 := PowerSeriesRing(Integers():Precision := 25); %o A298802 FG := FreeGroup(2); %o A298802 TG := quo; %o A298802 f, A :=IsAutomaticGroup(TG); %o A298802 gf := GrowthFunction(A); %o A298802 R!gf; %o A298802 Coefficients(PSR25!gf); %o A298802 (PARI) Vec((1 + x)^2*(1 + x^2) / (1 - 2*x - 2*x^3 + x^4) + O(x^40)) \\ _Colin Barker_, Feb 04 2018 %Y A298802 Cf. A008579. %K A298802 nonn,easy,new %O A298802 0,2 %A A298802 _John Cannon_ and _N. J. A. Sloane_, Feb 04 2018 %I A299159 %S A299159 1,2,5,15,32,42,60,110,120,152,215,242,260,315,357,390,392,425,470, %T A299159 495,560,732,735,840,1055,1082,1127,1275,1307,1352,1457,1562,1590, %U A299159 1755,1782,1797,1887,1925,1967,2055,2112,2132,2150,2175,2205,2360,2387,2472,2517,2567,2667,2717,2822,2882,2930,2945 %N A299159 Numbers k such that 4*k-1, 6*k-1 and 12*k-1 are prime. %H A299159 Robert Israel, Table of n, a(n) for n = 1..10000 %p A299159 select(j -> isprime(4*j-1) and isprime(6*j-1) and isprime(12*j-1), [$1..10000]); %t A299159 Select[Range[3000], And@@PrimeQ/@({4, 6, 12} # - 1) &] (* _Vincenzo Librandi_, Feb 04 2018 *) %o A299159 (MAGMA) [n: n in [0..3000] |IsPrime(4*n-1) and IsPrime(6*n-1) and IsPrime(12*n-1)]; // _Vincenzo Librandi_, Feb 04 2018 %Y A299159 Cf. A299068. %K A299159 nonn,new %O A299159 1,2 %A A299159 _Robert Israel_, Feb 04 2018 %I A299155 %S A299155 329400,175472640,808214400,1367566200,1928871000,3433706640 %N A299155 Numbers i such that Fibonacci(i) is divisible by i+k for k=0..6. %C A299155 A subsequence of A298687. %Y A299155 Cf. A000045, A023172, A217738, A221018, A225219, A298684, A298685, A298686, A298687. %K A299155 nonn,more,new %O A299155 1,1 %A A299155 _Chai Wah Wu_, Feb 03 2018 %I A297777 %S A297777 1,1,1,1,1,1,1,1,2,1,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,1,2,2,2,2, %T A297777 2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2, %U A297777 2,1,2,2,2,2,2,2,2,2,2,1,2,2,3,3,3,3 %N A297777 Number of distinct runs in base-9 digits of n. %C A297777 Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. %H A297777 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297777 6643 in base-9: 1,0,1,0,1; five runs, of which 2 are distinct, so that a(6643) = 2. %t A297777 b = 9; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297777 Table[s[n], {n, 1, 200}] %Y A297777 Cf. A043561 (number of runs, not necessarily distinct), A297770. %K A297777 nonn,base,easy,new %O A297777 1,9 %A A297777 _Clark Kimberling_, Jan 29 2018 %I A297774 %S A297774 1,1,1,1,1,2,1,2,2,2,2,2,2,1,2,2,2,2,2,2,1,2,2,2,2,2,2,1,2,2,2,2,2,2, %T A297774 1,2,2,3,3,3,3,2,1,2,2,2,2,3,2,2,3,3,3,3,2,3,2,3,3,3,2,3,3,2,3,3,2,3, %U A297774 3,3,2,2,3,2,3,3,3,3,2,2,3,3,3,2,2,1 %N A297774 Number of distinct runs in base-6 digits of n. %C A297774 Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. %H A297774 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297774 9^9 in base-6: 1,0,2,2,3,5,4,3,3,2,1,3; ten runs, of which 8 are distinct, so that a(9^9) = 8. %t A297774 b = 6; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297774 Table[s[n], {n, 1, 200}] %Y A297774 Cf. A043558 (number of runs, not necessarily distinct), A297770. %K A297774 nonn,base,easy,new %O A297774 1,6 %A A297774 _Clark Kimberling_, Jan 27 2018 %I A297556 %S A297556 1,6,7,19,99,115,307,1587,1843,4915,25395,29491,78643,406323,471859, %T A297556 1258291,6501171,7549747,20132659,104018739,120795955,322122547, %U A297556 1664299827,1932735283,5153960755,26628797235,30923764531,82463372083,426060755763,494780232499 %N A297556 a(n) = a(n-1) + 16*a(n-3) - 16*a(n-4), where a(0) = 1, a(1) = 4, a(2) = 7, a(3) = 19. %C A297556 Conjecture: a(n) = least positive whose base-4 up-variation is n; see column 1 of A297553. %H A297556 Clark Kimberling, Table of n, a(n) for n = 0..1000 %H A297556 Index entries for linear recurrences with constant coefficients, signature (1,0,16,-16) %F A297556 a(n) = a(n-1) + 16*a(n-3) - 16*a(n-4), where a(0) = 1, a(1) = 4, a(2) = 7, a(3) = 19. %F A297556 G.f.: (1 + 5 x + x^2 - 4 x^3)/(1 - x - 16 x^3 + 16 x^4). %t A297556 LinearRecurrence[{1, 0, 16, -16}, {1, 6, 7, 19}, 40] %Y A297556 Cf. A297553. %K A297556 nonn,easy,new %O A297556 0,2 %A A297556 _Clark Kimberling_, Jan 21 2018 %I A297555 %S A297555 1,4,8,12,76,140,204,1228,2252,3276,19660,36044,52428,314572,576716, %T A297555 838860,5033164,9227468,13421772,80530636,147639500,214748364, %U A297555 1288490188,2362232012,3435973836,20615843020,37795712204,54975581388,329853488332,604731395276 %N A297555 a(n) = a(n-1) + 16*a(n-3) - 16*a(n-4), where a(0) = 1, a(1) = 4, a(2) = 8, a(3) = 12, a(4) = 76. %C A297555 Conjecture: a(n) = least positive whose base-4 down-variation is n; see column 1 of A297552. %H A297555 Clark Kimberling, Table of n, a(n) for n = 0..1000 %H A297555 Index entries for linear recurrences with constant coefficients, signature (1,0,16,-16) %F A297555 a(n) = a(n-1) + 16*a(n-3) - 16*a(n-4), where a(0) = 1, a(1) = 4, a(2) = 8, a(3) = 12, a(4) = 76. %F A297555 G.f.: (1 + 3 x + 4 x^2 - 12 x^3 + 16 x^4)/(1 - x - 16 x^3 + 16 x^4) %t A297555 Join[{1}, LinearRecurrence[{1, 0, 16, -16}, {4, 8, 12, 76}, 40]] %Y A297555 Cf. A297552. %K A297555 nonn,easy,new %O A297555 0,2 %A A297555 _Clark Kimberling_, Jan 21 2018 %I A297443 %S A297443 1,3,6,11,20,33,60,101,182,303,546,911,1640,2733,4920,8201,14762, %T A297443 24603,44286,73811,132860,221433,398580,664301,1195742,1992903, %U A297443 3587226,5978711,10761680,17936133,32285040,53808401,96855122,161425203,290565366,484275611 %N A297443 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - 3*a(n-5), where a(0) = 1, a(1) = 3, a(2) = 6, a(3) = 11, a(4) = 20, a(5) = 33. %C A297443 Conjecture: a(n) = least positive whose base-3 total variation is n; see A297440. %H A297443 Clark Kimberling, Table of n, a(n) for n = 0..1000 %H A297443 Index entries for linear recurrences with constant coefficients, signature (1,2,-2,3,-3) %F A297443 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - 3*a(n-5), where a(0) = 1, a(1) = 3, a(2) = 6, a(3) = 11, a(4) = 20, a(5) = 33. %F A297443 G.f.: (1 + 2*x + x^2 + x^3 - 3*x^5) / ((1 - x)*(1 + x^2)*(1 - 3*x^2)). - Corrected by _Colin Barker_, Jan 21 2018 %t A297443 Join[{1}, LinearRecurrence[{1, 2, -2, 3, -3}, {3, 6, 11, 20, 33}, 40]] %o A297443 (PARI) Vec((1 + 2*x + x^2 + x^3 - 3*x^5) / ((1 - x)*(1 + x^2)*(1 - 3*x^2)) + O(x^40)) \\ _Colin Barker_, Jan 21 2018 %Y A297443 Cf. A297440. %K A297443 nonn,easy,new %O A297443 0,2 %A A297443 _Clark Kimberling_, Jan 21 2018 %I A297442 %S A297442 1,2,5,3,10,11,4,14,20,47,6,15,29,92,101,7,16,32,100,182,425,8,17,33, %T A297442 104,263,830,911,9,19,34,128,290,902,1640,3827,12,23,35,137,299,910, %U A297442 2369,7472,8201,13,28,38,140,302,914,2612,8120,14762,34445 %N A297442 Rectangular array R by antidiagonals: row n shows the positive integers whose base-3 digits have up-variation n, for n>=0. See Comments. %C A297442 Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See A297330 for a guide to related sequences and partitions of the natural numbers. %C A297442 Every positive integer occurs exactly once in the array, so that as a sequence this is a permutation of the positive integers. %C A297442 Conjecture: each column, after some number of initial terms, satisfies the linear recurrence relation c(n) = c(n-1) + 9*c(n-2) - 9*c(n-3). %e A297442 Northwest corner: %e A297442 1 2 3 4 6 7 8 9 %e A297442 5 10 14 15 16 17 19 23 %e A297442 11 20 29 32 33 34 35 38 %e A297442 47 92 100 104 128 137 140 141 %e A297442 101 182 263 290 299 302 303 304 %e A297442 425 830 902 910 914 938 1154 1235 %t A297442 g[n_, b_] := Differences[IntegerDigits[n, b]]; %t A297442 b = 3; z = 200000; u = Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}] ; %t A297442 p[n_] := Position[u, n]; TableForm[Table[Take[Flatten[p[n]], 15], {n, 0, 9}]] %t A297442 v[n_, k_] := p[k - 1][[n]]; %t A297442 w = Table[v[k, n - k + 1], {n, 10}, {k, n, 1, -1}] // Flatten %Y A297442 Cf. A297445 (conjectured 1st column), A297440, A297441. %K A297442 nonn,tabl,easy,new %O A297442 1,2 %A A297442 _Clark Kimberling_, Jan 21 2018 %I A297330 %S A297330 0,0,0,0,0,0,0,0,0,1,0,1,2,3,4,5,6,7,8,2,1,0,1,2,3,4,5,6,7,3,2,1,0,1, %T A297330 2,3,4,5,6,4,3,2,1,0,1,2,3,4,5,5,4,3,2,1,0,1,2,3,4,6,5,4,3,2,1,0,1,2, %U A297330 3,7,6,5,4,3,2,1,0,1,2,8,7,6,5,4,3,2 %N A297330 Total variation of base-10 digits of n; see Comments. %C A297330 Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). Guide to related sequences and partitions of the natural numbers: %C A297330 *** %C A297330 Base b {DV(n,b)} {UV(n,b)} {TV(n,b)} %C A297330 2 A033264 A037800 A037834 %C A297330 3 A037853 A037844 A037835 %C A297330 4 A037854 A037845 A037836 %C A297330 5 A037855 A037846 A037837 %C A297330 6 A037856 A037847 A037838 %C A297330 7 A037857 A037848 A037839 %C A297330 8 A037858 A037849 A037840 %C A297330 9 A037859 A037850 A037841 %C A297330 10 A037860 A037851 A297330 %C A297330 11 A297231 A297232 A297233 %C A297330 12 A297234 A297235 A297236 %C A297330 13 A297237 A297238 A297239 %C A297330 14 A297240 A297241 A297242 %C A297330 15 A297243 A297244 A297245 %C A297330 16 A297246 A297247 A297247 %C A297330 For each b, let u = {n : UV(n,b) < DV(n,b)}, e = {n : UV(n,b) = DV(n,b)}, and d = {n : UV(n,b) > DV(n,b)}. The sets u,e,d partition the natural numbers. A guide to the matching sequences for u, e, d follows: %C A297330 *** %C A297330 Base b Sequence u Sequence e Sequence d %C A297330 2 A005843 A005408 (none) %C A297330 3 A297249 A297250 A297251 %C A297330 4 A297252 A297253 A297254 %C A297330 5 A297255 A297256 A297257 %C A297330 6 A297258 A297259 A297260 %C A297330 7 A297261 A297262 A297263 %C A297330 8 A297264 A297265 A297266 %C A297330 9 A297267 A297268 A297269 %C A297330 10 A297270 A297271 A297272 %C A297330 11 A297273 A297274 A297275 %C A297330 12 A297276 A297277 A297278 %C A297330 13 A297279 A297280 A297281 %C A297330 14 A297282 A297283 A297284 %C A297330 15 A297285 A297286 A297287 %C A297330 16 A297288 A297289 A297290 %C A297330 Not a duplicate of A151950: e.g., a(100)=1 but A151950(100)=11. - _Robert Israel_, Feb 06 2018 %H A297330 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297330 13684632 has DV = 8-4 + 6-3 + 3-2 = 8 and has UV = 3-1 + 6-3 + 8-6 + 6-4 = 9, so that a(13684632) = DV + UV = 17. %p A297330 f:= proc(n) local L,i; L:= convert(n,base,10); %p A297330 add(abs(L[i+1]-L[i]),i=1..nops(L)-1) end proc: %p A297330 map(f, [$1..100]); # _Robert Israel_, Feb 04 2018 %t A297330 b = 10; z = 120; t = Table[Total@Flatten@Map[Abs@Differences@# &, Partition[IntegerDigits[n, b], 2, 1]], {n, z}] (* after _Michael De Vlieger_, e.g. A037834 *) %Y A297330 Cf. A037851, A297330, A297271, A297272. %K A297330 nonn,base,easy,new %O A297330 1,13 %A A297330 _Clark Kimberling_, Jan 17 2018 %I A297247 %S A297247 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,0, %T A297247 0,0,1,2,3,4,5,6,7,8,9,10,11,12,13,0,0,0,0,1,2,3,4,5,6,7,8,9,10,11,12, %U A297247 0,0,0,0,0,1,2,3,4,5,6,7,8,9,10,11,0,0,0 %N A297247 Up-variation of the base-16 digits of n; see Comments. %C A297247 Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). Every positive integer occurs infinitely many times. See A297330 for a guide to related sequences and partitions of the natural numbers. %H A297247 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297247 20 in base 16: 1,4; here UV = 3, so that a(20) = 3. %t A297247 g[n_, b_] := Differences[IntegerDigits[n, b]]; %t A297247 b = 16; z = 120; Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}]; (* A297246 *) %t A297247 Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}]; (* A297247 *) %Y A297247 Cf. A297246, A297248, A297330. %K A297247 nonn,base,easy,new %O A297247 1,19 %A A297247 _Clark Kimberling_, Jan 18 2018 %I A296908 %S A296908 3720,3721,3780,3781,3782,3840,3841,3842,3843,3900,3901,3902,3903, %T A296908 3904,3960,3961,3962,3963,3964,3965,4020,4021,4022,4023,4024,4025, %U A296908 4026,4080,4081,4082,4083,4084,4085,4086,4087,4140,4141,4142,4143,4144,4145,4146,4147 %N A296908 Numbers n whose base-60 digits d(m), d(m-1), ..., d(0) have #(pits) < #(peaks); see Comments. %C A296908 A pit is an index i such that d(i-1) > d(i) < d(i+1); a peak is an index i such that d(i-1) < d(i) > d(i+1). The sequences A296906..A296908 partition the natural numbers. We have a(n) = A000027(n) for n=1..3600, but not for n = 3601. See the guides at A296712 and A296882. %H A296908 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A296908 The base-60 digits of 13395721 are 1,2,1,2,1; here #(pits) = 1 and #(peaks) = 2, so that 13395721 is in the sequence. %t A296908 z = 200; b = 60; %t A296908 d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]]; %t A296908 Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &] (* A296906 *) %t A296908 Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &] (* A296907 *) %t A296908 Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &] (* A296908 *) %Y A296908 Cf. A296882, A296712, A296906, A296907. %K A296908 nonn,base,easy,new %O A296908 1,1 %A A296908 _Clark Kimberling_, Jan 12 2018 %I A296907 %S A296907 3601,3602,3603,3604,3605,3606,3607,3608,3609,3610,3611,3612,3613, %T A296907 3614,3615,3616,3617,3618,3619,3620,3621,3622,3623,3624,3625,3626, %U A296907 3627,3628,3629,3630,3631,3632,3633,3634,3635,3636,3637,3638,3639,3640,3641,3642,3643 %N A296907 Numbers n whose base-60 digits d(m), d(m-1), ..., d(0) have #(pits) > #(peaks); see Comments. %C A296907 A pit is an index i such that d(i-1) > d(i) < d(i+1); a peak is an index i such that d(i-1) < d(i) > d(i+1). The sequences A296906..A296908 partition the natural numbers. We have a(n) = A000027(n) for n=1..3600, but not for n = 3601. See the guides at A296712 and A296882. %H A296907 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A296907 The base-60 digits of 26143262 are 2,1,2,1,2; here #(pits) = 2 and #(peaks) = 1, so that 26143262 is in the sequence. %t A296907 z = 200; b = 60; %t A296907 d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]]; %t A296907 Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &] (* A296906 *) %t A296907 Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &] (* A296907 *) %t A296907 Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &] (* A296908 *) %Y A296907 Cf. A296882, A296712, A296906, A296908. %K A296907 nonn,base,easy,new %O A296907 1,1 %A A296907 _Clark Kimberling_, Jan 12 2018 %I A296906 %S A296906 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26, %T A296906 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49, %U A296906 50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67 %N A296906 Numbers n whose base-60 digits d(m), d(m-1), ..., d(0) have #(pits) = #(peaks); see Comments. %C A296906 A pit is an index i such that d(i-1) > d(i) < d(i+1); a peak is an index i such that d(i-1) < d(i) > d(i+1). The sequences A296906..A296908 partition the natural numbers. We have a(n) = A000027(n) for n=1..3600, but not for n = 3601. See the guides at A296712 and A296882. %H A296906 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A296906 The base-60 digits of 223262 are 1,2,1,2; here #(pits) = 1 and #(peaks) = 1, so that 223262 is in the sequence. %t A296906 z = 200; b = 60; %t A296906 d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]]; %t A296906 Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &] (* A296906 *) %t A296906 Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &] (* A296907 *) %t A296906 Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &] (* A296908 *) %Y A296906 Cf. A296882, A296712, A296907, A296908. %K A296906 nonn,base,easy,new %O A296906 1,2 %A A296906 _Clark Kimberling_, Jan 12 2018 %I A296904 %S A296904 401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417, %T A296904 418,419,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815, %U A296904 816,817,818,819,822,823,824,825,826,827,828,829,830,831,832,833,834,835 %N A296904 Numbers n whose base-20 digits d(m), d(m-1), ..., d(0) have #(pits) > #(peaks); see Comments. %C A296904 A pit is an index i such that d(i-1) > d(i) < d(i+1); a peak is an index i such that d(i-1) < d(i) > d(i+1). The sequences A296903..A296905 partition the natural numbers. See the guides at A296712 and A296882. %H A296904 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A296904 The base-20 digits of 328822 are 2,1,2,1,2; here #(pits) = 2 and #(peaks) = 1, so that 328822 is in the sequence. %t A296904 z = 200; b = 20; %t A296904 d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]]; %t A296904 Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &] (* A296903 *) %t A296904 Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &] (* A296904 *) %t A296904 Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &] (* A296905 *) %Y A296904 Cf. A296882, A296712, A296904, A296905. %K A296904 nonn,base,easy,new %O A296904 1,1 %A A296904 _Clark Kimberling_, Jan 12 2018 %I A298694 %S A298694 1,1,4,32,419,8052,207784,6724274,260396693,11697865930,596886780272, %T A298694 34072732137625,2151062784054901,148819021611467291, %U A298694 11198412956841549966,910736443741061568539,79616310026220269203631,7446056807577515910468813,741918566779386113373532994,78467177619239380045368550016,8779922184077661414128958823323 %N A298694 G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n*(n+1), n)/(n+1) * x^n / A(x)^(n^2). %e A298694 G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 419*x^4 + 8052*x^5 + 207784*x^6 + 6724274*x^7 + 260396693*x^8 + 11697865930*x^9 + 596886780272*x^10 + 34072732137625*x^11 + 2151062784054901*x^12 + 148819021611467291*x^13 + 11198412956841549966*x^14 + 910736443741061568539*x^15 + ... %e A298694 such that %e A298694 A(x) = 1 + C(2,1)/2*x/A(x) + C(6,2)/3*x^2/A(x)^4 + C(12,3)/4*x^3/A(x)^9 + C(20,4)/5*x^4/A(x)^16 + C(30,5)/6*x^5/A(x)^25 + C(42,6)/7*x^6/A(x)^36 + C(56,7)/8*x^7/A(x)^49 + ... %e A298694 more explicitly, %e A298694 A(x) = 1 + x/A(x) + 5*x^2/A(x)^4 + 55*x^3/A(x)^9 + 969*x^4/A(x)^16 + 23751*x^5/A(x)^25 + 749398*x^6/A(x)^36 + 28989675*x^7/A(x)^49 + ... + A135861(n)*x^n/A(x)^(n^2) + ... %o A298694 (PARI) {a(n) = my(A=[1]); for(i=1, n, A = Vec(sum(m=0, #A, binomial(m*(m+1), m)/(m+1) * x^m/Ser(A)^(m^2) ))); A[n+1]} %o A298694 for(n=0,20,print1(a(n),", ")) %Y A298694 Cf. A298691, A298692, A298693, A135861. %K A298694 nonn,new %O A298694 0,3 %A A298694 _Paul D. Hanna_, Feb 03 2018 %I A298693 %S A298693 1,1,3,22,294,5911,158293,5251690,206696194,9387611937,482745371458, %T A298693 27717788095397,1757818683339028,122058148921357056, %U A298693 9212494564360610855,751138761646263512978,65807775099574132000968,6166278653572358495161057,615421469545011786309942067,65183859793912213778457542207,7303117991652113167690085149033 %N A298693 G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n*(n+1), n)/(n+1) * x^n / A(x)^( n*(n+1) ). %e A298693 G.f.: A(x) = 1 + x + 3*x^2 + 22*x^3 + 294*x^4 + 5911*x^5 + 158293*x^6 + 5251690*x^7 + 206696194*x^8 + 9387611937*x^9 + 482745371458*x^10 + 27717788095397*x^11 + 1757818683339028*x^12 + 122058148921357056*x^13 + 9212494564360610855*x^14 + 751138761646263512978*x^15 + ... %e A298693 such that %e A298693 A(x) = 1 + C(2,1)/2*x/A(x)^2 + C(6,2)/3*x^2/A(x)^6 + C(12,3)/4*x^3/A(x)^12 + C(20,4)/5*x^4/A(x)^20 + C(30,5)/6*x^5/A(x)^30 + C(42,6)/7*x^6/A(x)^42 + C(56,7)/8*x^7/A(x)^56 + ... %e A298693 more explicitly, %e A298693 A(x) = 1 + x/A(x)^2 + 5*x^2/A(x)^6 + 55*x^3/A(x)^12 + 969*x^4/A(x)^20 + 23751*x^5/A(x)^30 + 749398*x^6/A(x)^42 + 28989675*x^7/A(x)^56 + ... + A135861(n)*x^n/A(x)^(n*(n+1)) + ... %o A298693 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = Vec(sum(m=0,#A,binomial(m*(m+1),m)/(m+1) * x^m/Ser(A)^(m*(m+1)) ))); A[n+1]} %o A298693 for(n=0,20,print1(a(n),", ")) %Y A298693 Cf. A298692, A135861. %K A298693 nonn,new %O A298693 0,3 %A A298693 _Paul D. Hanna_, Feb 03 2018 %I A298692 %S A298692 1,1,2,15,213,4485,123566,4171778,166069875,7602292250,393220294258, %T A298692 22679300697606,1443478702575162,100529312696403699, %U A298692 7606562231567559478,621526322941129712986,54553240678513466719077,5120001583257750960650134,511729676123794537164792892,54270040973557127212080028474,6087267497390906756985330494931 %N A298692 G.f. A(x) satisfies: 1 = Sum_{n>=0} binomial( n*(n+1), n)/(n+1) * x^n / A(x)^( (n+1)^2 ). %C A298692 Compare to: 1 = Sum_{n>=0} binomial(m*(n+1), n)/(n+1) * x^n / (1+x)^(m*(n+1)) holds for fixed m. %H A298692 Paul D. Hanna, Table of n, a(n) for n = 0..100 %e A298692 G.f.: A(x) = 1 + x + 2*x^2 + 15*x^3 + 213*x^4 + 4485*x^5 + 123566*x^6 + 4171778*x^7 + 166069875*x^8 + 7602292250*x^9 + 393220294258*x^10 + 22679300697606*x^11 + 1443478702575162*x^12 + 100529312696403699*x^13 + 7606562231567559478*x^14 + 621526322941129712986*x^15 + ... %e A298692 such that %e A298692 1 = 1/A(x) + C(2,1)/2*x/A(x)^4 + C(6,2)/3*x^2/A(x)^9 + C(12,3)/4*x^3/A(x)^16 + C(20,4)/5*x^4/A(x)^25 + C(30,5)/6*x^5/A(x)^36 + C(42,6)/7*x^6/A(x)^49 + C(56,7)/8*x^7/A(x)^64 + ... %e A298692 more explicitly, %e A298692 1 = 1/A(x) + x/A(x)^4 + 5*x^2/A(x)^9 + 55*x^3/A(x)^16 + 969*x^4/A(x)^25 + 23751*x^5/A(x)^36 + 749398*x^6/A(x)^49 + 28989675*x^7/A(x)^64 + ... + A135861(n)*x^n/A(x)^((n+1)^2) + ... %t A298692 terms = 21; A[_] = 1; Do[A[x_] = A[x] - 1 + Sum[Binomial[n*(n+1), n]/(n+1)*x^n/A[x]^((n + 1)^2) + O[x]^(terms), {n, 0, k}], {k, terms}]; CoefficientList[A[x], x] (* _Jean-François Alcover_, Feb 06 2018 *) %o A298692 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = Vec(sum(m=0,#A,binomial(m*(m+1),m)/(m+1) * x^m/Ser(A)^((m+1)^2-1) ))); A[n+1]} %o A298692 for(n=0,20,print1(a(n),", ")) %Y A298692 Cf. A135861. %K A298692 nonn,new %O A298692 0,3 %A A298692 _Paul D. Hanna_, Feb 03 2018 %I A296307 %S A296307 1,5,1,11,2,1,19,7,2,1,29,9,3,2,1,41,17,9,3,2,1,55,20,11,4,3,2,1,71, %T A296307 31,13,11,4,3,2,1,89,35,23,13,5,4,3,2,1,109,49,26,15,13,5,4,3,2,1,131, %U A296307 54,29,17,15,6,5,4,3,2,1,155,71,43,29,17,15,6,5,4,3 %N A296307 Array read by upwards antidiagonals: f(n,k) = (n+1)*ceiling(n/(k-1)) - 1. %C A296307 f(n,k) = (n+1)*ceiling(n/(k-1))-1 is the Frobenius number F(n+1,n+2,...,n+k), k>1. This formula is derived in "Frobenius number for a set of successive numbers". %C A296307 f(n,k) is the greatest number which is not a linear combination of n+1,n+2,...,n+k with nonnegative coefficients. %C A296307 Example: f(2,3) = 5 because 6=2*3, 7=3+4, 8=2*4, 9=3*3, 10=2*3+4 and so on. %C A296307 Special sequences: f(n,2) = A028387(n), f(n,3) = A079326(n+1), f(n,4) = A138984(n), f(n,5) = A138985(n), f(n,6) = A138986(n), f(n,7) = A138987(n), f(n,8) = A138988(n). %C A296307 f(n,k) is a generalization of these sequences. %H A296307 Gerhard Kirchner, Table of n, a(n) for n = 1..1000 %H A296307 Gerhard Kirchner, Frobenius number for a set of successive numbers %H A296307 Gerhard Kirchner, Table of Frobenius numbers %H A296307 Gerhard Kirchner, Table of Frobenius numbers %e A296307 Example: %e A296307 f(n,2) f(n,3) f(n,4) %e A296307 a(1)= 1 a(3)=1 a(6) =1 %e A296307 a(2)= 5 a(5)=2 a(9) =2 %e A296307 a(4)=11 a(8)=7 a(13)=3 %e A296307 More terms in "Table of Frobenius numbers". %Y A296307 Cf. A028387, A079326, A138984, A138985, A138986, A138987, A138988. %K A296307 nonn,tabl,new %O A296307 1,2 %A A296307 _Gerhard Kirchner_, Dec 10 2017 %I A297707 %S A297707 1,2,18,768,90000,44789760,30494620800,121762322841600, %T A297707 393644011735296000,5618427494400000000000, %U A297707 107587910030480590233600000,5951222311476064581656248320000,176804782652901880753915871232000000 %N A297707 a(n) = Product_{k=1..n-1} n!k for n>1, a(1) = 1, where n!k is k-tuple factorial of n. %C A297707 What is the least n > 2 for which a(n) - prevprime(a(n)) is a composite number? If such a number n exists, it is greater than 250. %C A297707 The least n for which nextprime(a(n)) - a(n) is a composite number is 158. %F A297707 a(n) = Product_{t=1..n-1} (Product_{k=0..floor((n-1)/t)} (n-t*k)) for n>1. %F A297707 a(n) = (n^(n-1))*Product_{k=1..n-1} k^tau(n-k) for n>1. %e A297707 a(2) = (2!1) = (2*1) = 2; %e A297707 a(3) = (3!1)*(3!2) = (3*2*1)*(3*1) = 18; %e A297707 a(4) = (4!1)*(4!2)*(4!3) = (4*3*2*1)*(4*2)*(4*1) = 768; %e A297707 a(5) = (5!1)*(5!2)*(5!3)*(5!4) = (5*4*3*2*1)*(5*3*1)*(5*2)*(5*1) = 90000. %t A297707 Array[(#^(# - 1)) Product[k^DivisorSigma[0, # - k], {k, # - 1}] &, 13] (* _Michael De Vlieger_, Jan 04 2018 *) %Y A297707 Cf. A000142, A006882, A007661, A007662, A085157, A085158, A114799, A114800. %Y A297707 Cf. A114806, A288327, A006990, A033933. %K A297707 nonn,new %O A297707 1,2 %A A297707 _Lechoslaw Ratajczak_, Jan 03 2018 %I A295078 %S A295078 6,28,40,84,120,140,224,234,270,420,468,496,672,756,936,1080,1120, %T A295078 1170,1372,1488,1550,1638,1782,1862,2176,2340,2480,2574,3100,3250, %U A295078 3276,3360,3472,3564,3724,3744,3780,4116,4464,4598,4650,4680,5148,5456,5586,6048,6200 %N A295078 Numbers n > 1 such that n and sigma(n) have the same smallest and simultaneously largest prime factors. %C A295078 All even perfect numbers are terms. %C A295078 Conjecture: A007691 (multiply-perfect numbers) is a subsequence. %C A295078 Note that an odd perfect number (if it exists) would be a counterexample to the conjecture. - _Robert Israel_, Jan 08 2018 %C A295078 Intersection of A071834 and A295076. %C A295078 Numbers n such that A020639(n) = A020639(sigma(n)) and simultaneously A006530(n) = A006530(sigma(n)). %C A295078 Numbers n such that A020639(n) = A071189(n) and simultaneously A006530(n) = A071190(n). %C A295078 Supersequence of A027598. %H A295078 Jaroslav Krizek, Table of n, a(n) for n = 1..1000 %e A295078 40 = 2^3*5 and sigma(40) = 90 = 2*3^2*5 hence 40 is in the sequence. %e A295078 The first odd term is 29713401 = 3^2 * 23^2 * 79^2; sigma(29713401) = 45441669 = 3*7^3*13*43*79. %p A295078 filter:= proc(n) local f, s; uses numtheory; %p A295078 f:= factorset(n); %p A295078 s:= factorset(sigma(n)); %p A295078 min(f) = min(s) and max(f)=max(s) %p A295078 end proc: %p A295078 select(filter, [$2..10^4]); # _Robert Israel_, Jan 08 2018 %t A295078 Rest@ Select[Range@ 6200, SameQ @@ Map[{First@ #, Last@ #} &@ FactorInteger[#][[All, 1]] &, {#, DivisorSigma[1, #]}] &] (* _Michael De Vlieger_, Nov 13 2017 *) %o A295078 (MAGMA) [n: n in [2..10000] | Minimum(PrimeDivisors(n)) eq Minimum(PrimeDivisors(SumOfDivisors(n))) and Maximum(PrimeDivisors(n)) eq Maximum(PrimeDivisors(SumOfDivisors(n)))] %o A295078 (PARI) isok(n) = if (n > 1, my(fn = factor(n)[,1], fs = factor(sigma(n))[,1]); (vecmin(fn) == vecmin(fs)) && (vecmax(fn) == vecmax(fs))); \\ _Michel Marcus_, Jan 08 2018 %Y A295078 Cf. A000203, A006530, A007691, A020639, A027598, A071189, A071190, A295076. %K A295078 nonn,new %O A295078 1,1 %A A295078 _Jaroslav Krizek_, Nov 13 2017 %E A295078 Added condition n>1 to definition. Corrected b-file. - _N. J. A. Sloane_, Feb 03 2018 %I A298817 %S A298817 0,1,2,6,23,59,99,203,469,807,1615,3349,2266,4576,14042,25002,89193, %T A298817 131215,135904,814531,885682,60842,3969154,3370892,6742296,14350136, %U A298817 42766902,97565102,444197631 %N A298817 a(n) is the binary XOR of all n-bit prime numbers. %C A298817 XOR is the binary exclusive-or operator. %C A298817 a(1)=0 for compatibility with similar sequences, and because 0 and 1 are not primes. %C A298817 Note the sequence s(n)-a(n), where s(n)=A298816(n) is the binary XOR of all n-bit squares, begins: 1, -1, 2, 3, -14, -38, -87, -175, -20, -230, -1258, -2352, 3819, 9957, -1525, -9925, 31932, 21654, 264124, 226521, 405022, 2495526, 944510, 8579700, 15679080, 49342536, -35092149, -19209773, -131473914. The distribution of negative and positive terms does not look random: runs of negative terms are followed by runs of positive terms. %e A298817 There are two 4-bit primes, namely 11 and 13. a(4) = (11 XOR 13) = 6. %o A298817 (Python) %o A298817 from sympy import nextprime %o A298817 n = x = L = 2 %o A298817 print '0,', %o A298817 while L < 47: %o A298817 nextn = nextprime(n) %o A298817 if (nextn ^ n) > n: # if lengths of binary representations are different %o A298817 print str(x)+',', %o A298817 x = 0 %o A298817 prevL = L %o A298817 L = len(bin(nextn))-2 %o A298817 for j in range(prevL, L-1): print '0,', %o A298817 n = nextn %o A298817 x ^= n %o A298817 (PARI) a(n) = {my(x = 0); for (k=2^(n-1), 2^n-1, if (isprime(k), x = bitxor(x, k));); x;} \\ _Michel Marcus_, Jan 27 2018 %Y A298817 Cf. A000040, A007088, A070939, A035100, A298816. %Y A298817 Cf. also A014234, A104080. %K A298817 nonn,base,more,new %O A298817 1,3 %A A298817 _Alex Ratushnyak_, Jan 26 2018 %I A298816 %S A298816 1,0,4,9,9,21,12,28,449,577,357,997,6085,14533,12517,15077,121125, %T A298816 152869,400028,1041052,1290704,2556368,4913664,11950592,22421376, %U A298816 63692672,7674753,78355329,312723717,656197893,1089399836,2723474460,4196236289,2416016385,8186515468 %N A298816 a(n) is the binary XOR of all n-bit squares, with a(2)=0 indicating that no 2-bit squares exist. %C A298816 XOR is the binary exclusive-or operator. %e A298816 There are two squares such that their binary representation is 5 bits long, namely 16 and 25. a(5) = 9 because 25 XOR 16 = 9. %e A298816 There are four squares such that their binary representation is 7 bits long, namely 64, 81, 100 and 121. a(7) = (64 XOR 81 XOR 100 XOR 121) = 12. %o A298816 (Python) %o A298816 i = n = x = L = 1 %o A298816 while L < 47: %o A298816 i+=1 %o A298816 nextn = i*i %o A298816 if (nextn ^ n) > n: # if lengths of binary representations are different %o A298816 print str(x)+',', %o A298816 x = 0 %o A298816 prevL = L %o A298816 L = len(bin(nextn))-2 %o A298816 for j in range(prevL, L-1): print '0,', %o A298816 n = nextn %o A298816 x ^= n %Y A298816 Cf. A000290, A007088, A070939. %K A298816 nonn,base,new %O A298816 1,3 %A A298816 _Alex Ratushnyak_, Jan 26 2018 %I A298687 %S A298687 13440,19440,329400,600600,2499840,3150840,5590200,7660800,69069000, %T A298687 83980800,96049800,98385840,175472640,179663400,237484800,320498640, %U A298687 330663600,375396840,404351640,406380240,429660000,437940000,505234800,574585200,635980800 %N A298687 Numbers i such that Fibonacci(i) is divisible by i+k for k=0..5. %C A298687 A subsequence of A298686. %H A298687 Chai Wah Wu, Table of n, a(n) for n = 1..57 %o A298687 (Python) %o A298687 p0 = 0 %o A298687 p1 = 1 %o A298687 for i in xrange(1,1000000): %o A298687 if p1 % i == 0 and p1 % (i+1) == 0 and p1 % (i+2) == 0: %o A298687 if p1 % (i+3) == 0 and p1 % (i+4) == 0 and p1 % (i+5) == 0: print i %o A298687 p0, p1 = p1, p0+p1 %Y A298687 Cf. A000045, A023172, A217738, A221018, A225219, A298684, A298685, A298686. %K A298687 nonn,new %O A298687 1,1 %A A298687 _Alex Ratushnyak_, Jan 24 2018 %E A298687 a(9)-a(25) from _Chai Wah Wu_, Jan 27 2018 %I A298685 %S A298685 540,1200,1620,3060,5580,9180,9900,12600,13440,13680,18300,19440, %T A298685 19800,21000,24480,36900,43200,49680,50220,54120,57240,61560,65880, %U A298685 81180,83700,103680,104160,154080,155520,156060,156240,202440,229320,252000,279000,298200,302940 %N A298685 Numbers i such that Fibonacci(i) is divisible by i, i+1, i+2, and i+3. %C A298685 A subsequence of A298684. %t A298685 Select[Range[10^5], Function[{i, j}, AllTrue[i + Range[0, 3], Divisible[j, #] &]] @@ {#, Fibonacci@ #} &] (* _Michael De Vlieger_, Jan 28 2018 *) %o A298685 (PARI) isone(n, k) = !(fibonacci(n) % (n+k)); %o A298685 isok(n) = isone(n,0) && isone(n,1) && isone(n,2) && isone(n,3); \\ _Michel Marcus_, Jan 29 2018 %Y A298685 Cf. A000045, A023172, A217738, A221018, A225219, A298684. %K A298685 nonn,new %O A298685 1,1 %A A298685 _Alex Ratushnyak_, Jan 24 2018 %I A294369 %S A294369 0,1,2,4,8,10 %N A294369 Indices of Fibonacci numbers (A000045) that are triangular numbers (A000217). %C A294369 The sequence of Fibonacci numbers that are also triangular numbers begins: 0, 1, 1, 3, 21, 55. That is, 0 and 1 followed by A039595. %e A294369 Fibonacci(10)=55 is a triangular number, therefore 10 is in the sequence. %Y A294369 Cf. A000045, A000217, A039595, A162394, A121343, A121874. %K A294369 nonn,fini,full,new %O A294369 1,3 %A A294369 _Alex Ratushnyak_, Jan 24 2018 %I A298821 %S A298821 706866045116113,706866045126361,706866045126697,706866045126907, %T A298821 706866045128377,706866045128563,706866045128953,706866045129163, %U A298821 706866045129403,706866045130057,706866045130153,706866045130459,706866045130723,706866045130771,706866045131107,706866045155113,706866045155899,706866045156043,706866045156409,706866045156499 %N A298821 Primes p for which pi_{24,19}(p) - pi_{24,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m). %C A298821 This is a companion sequence to A298820 and the first discovered for pi_{24,19}(p) - pi_{24,1}(p) prime race. The full sequence up to 10^15 contains 5 sign-changing zones with 3436990 terms in total with A(3436990) = 766164822666883 as the last one. %H A298821 Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000 %H A298821 A. Granville, G. Martin, Prime Number Races, Amer. Math. Monthly 113 (2006), no. 1, 1-33. %H A298821 Richard H. Hudson, Carter Bays, The appearance of tens of billion of integers x with pi_{24, 13}(x) < pi_{24, 1}(x) in the vicinity of 10^12, Journal für die reine und angewandte Mathematik, 299/300 (1978), 234-237. MR 57 #12418. %H A298821 M. Rubinstein, P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173-197. %H A298821 Eric Weisstein's World of Mathematics, Prime Quadratic Effect. %Y A298821 Cf. A295355, A295356, A297449, A297450 %K A298821 nonn,new %O A298821 1,1 %A A298821 Andrey S. Shchebetov and _Sergei D. Shchebetov_, Jan 27 2018 %I A298820 %S A298820 21317046795798,21317046796093,21317046796102,21317046796104, %T A298820 21317046796154,21317046796159,21317046796172,21317046796185, %U A298820 21317046796193,21317046796208,21317046796212,21317046796221,21317046796226,21317046796229,21317046796240,21317046796968,21317046796986,21317046796992,21317046797002,21317046797007 %N A298820 Values of n for which pi_{24,19}(p_n) - pi_{24,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m). %C A298820 This is a companion sequence to A298821 and the first discovered for pi_{24,19}(p) - pi_{24,1}(p) prime race. The full sequence up to 10^15 contains 5 sign-changing zones with 3436990 terms in total with A(3436990) = 23049274819456 as the last one. %H A298820 Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000 %H A298820 A. Granville, G. Martin, Prime Number Races, Amer. Math. Monthly 113 (2006), no. 1, 1-33. %H A298820 Richard H. Hudson, Carter Bays, The appearance of tens of billion of integers x with pi_{24, 13}(x) < pi_{24, 1}(x) in the vicinity of 10^12, Journal für die reine und angewandte Mathematik, 299/300 (1978), 234-237. MR 57 #12418. %H A298820 M. Rubinstein, P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173-197. %H A298820 Eric Weisstein's World of Mathematics, Prime Quadratic Effect. %Y A298820 Cf. A295355, A295356, A297449, A297450 %K A298820 nonn,new %O A298820 1,1 %A A298820 Andrey S. Shchebetov and _Sergei D. Shchebetov_, Jan 27 2018 %I A297450 %S A297450 617139273158713,617139273159121,617139273159337,617139273163729, %T A297450 617139273163793,617139273165889,617139273166121,617139273167057, %U A297450 617139273169273,617139273169513,617139273169729,617139273170137,617139273170401,617139273171217,617139273206009,617139273206993,617139273207449,617139273207929,617139273208001,617139273504913 %N A297450 Primes p for which pi_{24,17}(p) - pi_{24,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m). %C A297450 This is a companion sequence to A297449 and the first discovered for pi_{24,17}(p) - pi_{24,1}(p) prime race. The full sequence up to 10^15 contains 3 sign-changing zones with 963922 terms in total with A(963922) = 772739867710897 as the last one. %H A297450 Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000 %H A297450 A. Granville, G. Martin, Prime Number Races, Amer. Math. Monthly 113 (2006), no. 1, 1-33. %H A297450 Richard H. Hudson, Carter Bays, The appearance of tens of billion of integers x with pi_{24, 13}(x) < pi_{24, 1}(x) in the vicinity of 10^12, Journal für die reine und angewandte Mathematik, 299/300 (1978), 234-237. MR 57 #12418. %H A297450 M. Rubinstein, P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173-197. %H A297450 Eric Weisstein's World of Mathematics, Prime Quadratic Effect. %Y A297450 Cf. A295355, A295356. %K A297450 nonn,new %O A297450 1,1 %A A297450 Andrey S. Shchebetov and _Sergei D. Shchebetov_, Jan 27 2018 %I A297449 %S A297449 18687728175380,18687728175387,18687728175395,18687728175515, %T A297449 18687728175520,18687728175587,18687728175592,18687728175626, %U A297449 18687728175698,18687728175707,18687728175715,18687728175726,18687728175738,18687728175762,18687728176789,18687728176820,18687728176831,18687728176844,18687728176846,18687728185530 %N A297449 Values of n for which pi_{24,17}(p_n) - pi_{24,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m). %C A297449 This is a companion sequence to A297450 and the first discovered for pi_{24,17}(p) - pi_{24,1}(p) prime race. The full sequence up to 10^15 contains 3 sign-changing zones with 963922 terms in total with A(963922) = 23241097440243 as the last one. %H A297449 Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000 %H A297449 A. Granville, G. Martin, Prime Number Races, Amer. Math. Monthly 113 (2006), no. 1, 1-33. %H A297449 Richard H. Hudson, Carter Bays, The appearance of tens of billion of integers x with pi_{24, 13}(x) < pi_{24, 1}(x) in the vicinity of 10^12, Journal für die reine und angewandte Mathematik, 299/300 (1978), 234-237. MR 57 #12418. %H A297449 M. Rubinstein, P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173-197. %H A297449 Eric Weisstein's World of Mathematics, Prime Quadratic Effect. %Y A297449 Cf. A295355, A295356. %K A297449 nonn,new %O A297449 1,1 %A A297449 Andrey S. Shchebetov and _Sergei D. Shchebetov_, Jan 27 2018 %I A298615 %S A298615 161,217,329,371,427,511,581,623,1246,791,1417,1243,1469,2071,917,973, %T A298615 1507,1529,1057,1099,1169,1211,1267,1969,1991,1393,2167,2189,2587, %U A298615 1477,2954,2321,2743,1631,1687,2629,2651,1757,1799,1841,1897,1981,3091,3113,2051,4102,3223,3809,7618,2219,4069,3487,4121,5947,2317,2359,4718,3707,2471,2513,5026,2569,2611,2681,2723,5446,2807,5161,2863,5726,4499,2947,4609,4631,3031,3101,3143,6286,3269,6019,5137,6071,3353,6706,3437,6331,3521,3563,7126,5599,6617,3661,5731,5753,6799,8857,8891,9937,3787,3829,7658,6017,3899,3941,3997,4039,8078,6347,4109,4151,4207,4249,4333 %N A298615 Let b(k) be A056240(k); this sequence lists numbers b(2n) such that there is at least one m>n for which b(2m)=3. A288313(n)=A056240(A298252(n)), A297150(n)=A056240(A297925(n)), and the terms of this sequence correspond to A056240(A298366). Distinct sequences A298252, A297925, and A298366 form a partition of the nonnegative even integers (A005843)>=6. These partitions holds because any even integer n>=6 is such that, either n-3 is prime (A298252), or n-5 is prime but n-3 is composite (A297925), or both n-5 and n-3 are composite (A298366). %F A298615 a(n) = A056240(A298366(n)). %e A298615 n=1, a(n)=A056240(A298366(1))=A056240(30)=161; n=24, a(24)=A056240(A298366(24))=A056240(190)=1969. %Y A298615 Cf. A056240, A288313, A297150, A298252, A298366, A005843. %K A298615 nonn,new %O A298615 1,1 %A A298615 _David James Sycamore_, Jan 26 2018 %I A298366 %S A298366 30,38,54,60,68,80,90,96,98,120,122,124,126,128,138,146,148,150,158, %T A298366 164,174,180,188,190,192,206,208,210,212,218,220,222,224,240,248,250, %U A298366 252,258,264,270,278,290,292,294,300,302,304,306,308,324,326,328,330,332,338,344,346,348,360,366,368,374,380,390,396,398,408,410,416,418,420,428,430,432,440,450,456,458,474,476,478,480,486,488,500,510,516,518,520,522,530,532,534,536,538,540,542,548,554,556,558,564,570,578,584,586,588,594,600 %N A298366 Even numbers n such that n-5 and n-3 are both composite. %C A298366 The sequence displays runs of consecutive even integers, whose frequency and length are related to gaps between successive primes local to these numbers. Where primes are rare (large gaps), the runs of consecutive even integers are longer (run length proportional to gap size). Let p=6. A string of r consecutive terms differing by 2 will start at p+7, and continue to q+1, where r=(g-4)/2. Thus at prime gap 8 a string of 2 consecutive terms differing by 2 will occur, at gap 10 there will be 3, and at gap 30 there will be 13; and so on. As the gap size increases by 2 so the run length of consecutive even terms increases by 1. The first occurrence of run length m occurs at the term corresponding to 7+A000230(m/2). %C A298366 The terms in this sequence, combined with those in A297925 and A298252 form a partition of A005843(n); n>=3; (nonnegative even numbers >=6). This is because any even integer n>=6 satisfies either: (i). n-3 is prime, (ii). n-5 is prime and n-3 is composite, or (iii). both n-5 and n-3 are composite. %C A298366 For any n>=1, A056240(a(n))=A298615(n). %e A298366 30 is included because 30-5=25, and 30-3=27; both composite, and 30 is the smallest even number with this property, hence a(1)=30. Also, A056240(a(1))=A056240(30)=161=A298615(1). 24 is not included because although 24-3=21, composite; 24-5 =19, prime. 210 is in this sequence, since 205 and 207 are both composite. 113 is the first prime to have a gap 14 ahead of it. Therefore we would expect a run of (14-4)/2=5 consecutive terms to start at 7+A000230(7)= 113+7=120; thus: 120,122,124,126,128. Likewise the first occurrence of run length 7 occurs at gap m=2*7+4=18, namely the term corresponding to 7+A000230(9)=523+7=530; thus: 530,532,534,536,538,540,542. %p A298366 N:=300: %p A298366 for n from 8 to N by 2 do %p A298366 if not isprime(n-5) and not isprime(n-3) then print(n); %p A298366 end if %p A298366 end do %Y A298366 Cf. A297925, A298252, A005843, A000230. %K A298366 nonn,new %O A298366 1,1 %A A298366 _David James Sycamore_, Jan 17 2018 %I A298252 %S A298252 6,8,10,14,16,20,22,26,32,34,40,44,46,50,56,62,64,70,74,76,82,86,92, %T A298252 100,104,106,110,112,116,130,134,140,142,152,154,160,166,170,176,182, %U A298252 184,194,196,200,202,214,226,230,232,236,242,244,254,260,266,272,274,280,284,286,296,310,314,316,320,334,340,350,352,356,362,370,376,382,386,392,400,404,412,422,424,434,436,442,446,452,460,464,466,470,482,490,494,502,506,512,524,526,544,550,560,566,572,574,580,590,602,604,610,616,620,622 %N A298252 Even integers n such that n-3 is prime. %C A298252 Subsequence of A005843, same as A113935 with first term (5) excluded, since it is odd, not even. Index in A056240 of terms in A288313 (except for first two terms 2,4 of latter). %C A298252 The terms in this sequence, combined with those in A297925 and A298366 form a partition of A005843(n); n>=3 (nonnegative numbers>=6). This is because any even integer n>=6 satisfies either(i) n-3 is prime, (ii) n-5 prime but n-3 composite, or (iii) n-5 and n-3 both composite. %F A298252 a(n)=A113935(n+1); n>=1. A056240(a(n))=A288313(n+2). %e A298252 a(1)=6 because 6-3=3; prime, and no smaller even number has this property; also a(1)=A113935(2)=6. a(2)=8 because 8-3=5 is prime; also A113935(3)=8. %e A298252 12 is not in the sequence because 12-3 = 9, composite. %p A298252 N:=200 %p A298252 for n from 6 to N by 2 do %p A298252 if isprime(n-3) then print(n); %p A298252 end if %p A298252 end do %t A298252 Select[2 Range@125, PrimeQ[# - 3] &] (* _Robert G. Wilson v_, Jan 15 2018 *) %Y A298252 Cf. A005843, A113935, A056240, A288313. %K A298252 nonn,new %O A298252 1,1 %A A298252 _David James Sycamore_, Jan 15 2018 %I A297150 %S A297150 35,65,95,115,155,185,215,235,265,305,335,365,395,415,445,485,515,545, %T A297150 565,635,655,695,755,785,815,835,865,905,965,995,1055,1115,1145,1165, %U A297150 1205,1255,1285,1315,1355,1385,1415,1465,1535,1565,1585,1655,1685,1745,1765,1795,1835,1865,1895,1915,1945,1985,2005,2045,2105,2165,2195,2215,2245,2285,2315,2335,2395,2435,2455,2495,2515,2545,2615,2705,2735,2785,2815,2855,2855,2885,2935,2965,3005,3035,3065,3095 %N A297150 Let b(k) denote A292081(k); the sequence lists numbers b(2n) where for all m > n, b(2m) > b(2n). %C A297150 This is also an ascending subsequence of the even indexed terms of A056240(2n) (of which A292081 is a subsequence). For n >= 1, a(n) is a semiprime of the form a(n)=5*A049591(n), and the index m in A056240 of any term in this sequence belongs to the sequence of even numbers m such that m-5 is prime and m-3 is not prime (A297925). See A297925 for explanation. %F A297150 a(n) = 5*A049591(n) = A056240(A297925(n)). %e A297150 a(1)=5*A049591(1)=5*7=35. Also A056240(A297925(1))=A056240(12)=35. %e A297150 a(17)=5*A049591(17)=5*103=515. Also A056240(A297925(17))=A056240(108)=515. %Y A297150 Similar to A288313. %Y A297150 Cf. A292081, A288313, A056240, A049591, A297925. %K A297150 nonn,new %O A297150 1,1 %A A297150 _David James Sycamore_, Dec 26 2017 %I A299110 %S A299110 7,41,239,9369319,63018038201,489133282872437279, %T A299110 19175002942688032928599, %U A299110 123426017006182806728593424683999798008235734137469123231828679 %N A299110 Probable primes in sequence {s_k(6)}, where s_k(6) = 6*s_{k-1}(6) - s_{k-2}(6), k >= 2, s_0(6) = 1, s_1(6) = 7. %C A299110 From a problem in A269254. %C A299110 Subsequent terms have too many digits to display. %C A299110 Appears to be a duplicate of A088165. - _R. J. Mathar_, Feb 06 2018 %F A299110 a(n) = s_{A299102(n)}(5) = A002315(A299102(n)). %Y A299110 Cf. A002315. %Y A299110 Cf. A299104, A299107, A299109, A299110, A117522, A299100-A299102. %Y A299110 Cf. A269253, A269254, A294099, A298675, A298677, A298678, A299045, A299071. %K A299110 nonn,hard,more,new %O A299110 1,1 %A A299110 _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Feb 02 2018 %I A299109 %S A299109 29,139,3191,15289,350981,1681651,20344613659,2237722536169, %T A299109 5650248517599839,1464318118372209903213451940281613111, %U A299109 471219735266432821374794400248484805597413615642086220363989152195627985749609 %N A299109 Probable primes in sequence {s_k(5)}, where s_k(5) = 5*s_{k-1}(5) - s_{k-2}(5), k >= 2, s_0(5) = 1, s_1(5) = 6. %C A299109 From a problem in A269254. %C A299109 Subsequent terms have too many digits to display. %F A299109 a(n) = s_{A299101(n)}(5) = A030221(A299101(n)). %Y A299109 Cf. A030221, A294099 (row 5). %Y A299109 Cf. A299104, A299107, A299109, A299110, A117522, A299100-A299102. %Y A299109 Cf. A269253, A269254, A298675, A298677, A298678, A299045, A299071. %K A299109 nonn,hard,more,new %O A299109 1,1 %A A299109 _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Feb 02 2018 %I A299107 %S A299107 5,19,71,3691,191861,138907099,26947261171, %T A299107 436315574686414344004975231616076636245689199862837798457639364993981991744926792179 %N A299107 Probable primes in sequence {s_k(4)}, where s_k(4) = 4*s_{k-1}(4) - s_{k-2}(4), k >= 2, s_0(4) = 1, s_1(4) = 5. %C A299107 From a problem in A269254. %C A299107 Subsequent terms have too many digits to display. %F A299107 a(n) = s_{A299100(n)}(4) = A001834(A299100(n)). %Y A299107 A269254, A001834, A294099 (row 4), %Y A299107 Cf. A285992, A299107, A299109, A299110, A117522, A299100-A299102. %Y A299107 Cf. A269253, A298675, A298677, A298678, A299045, A299071. %K A299107 nonn,hard,more,new %O A299107 1,1 %A A299107 _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Feb 02 2018 %I A299104 %S A299104 11,29,199,521,3571,9349,3010349,54018521,370248451,6643838879, %T A299104 119218851371,5600748293801,688846502588399,32361122672259149, %U A299104 412670427844921037470771,258899611203303418721656157249445530046830073044201152332257717521 %N A299104 Probable primes in sequence {s_k(3)}, where s_k(3) = 3*s_{k-1}(3) - s_{k-2}(3), k >= 2, s_0(3) = 1, s_1(3) = 4. %C A299104 From a problem in A269254. %C A299104 Subsequent terms have too many digits to display. %F A299104 a(n) = s_{A299099(n)}(3) = A002878(A299099(n)). %Y A299104 Cf. A269254, A002878, A294099 (row 3). %Y A299104 Cf. A299104, A299107, A299109, A299110, A299099--A299102. %Y A299104 Cf. A269253, A298675, A298677, A298678, A299045, A299071. %K A299104 nonn,hard,more,new %O A299104 1,1 %A A299104 _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Feb 02 2018 %I A299102 %S A299102 1,2,3,9,14,23,29,81,128,210,468,473,746,950,3344,4043,4839,14376, %T A299102 39521 %N A299102 Indices k such that s_k(6) is a (probable) prime, where s_k(6) = 6*s_{k-1}(6) - s_{k-2}(6), k >= 2, s_0(6) = 1, s_1(6) = 7. %C A299102 From a problem in A269254. %C A299102 Looks like a duplicate of A113501. - _R. J. Mathar_, Feb 06 2018 %Y A299102 Cf. A002315, A294099 (row 6). %Y A299102 Cf. A285992, A299107, A299109, A299110, A117522, A299100-A299102. %Y A299102 Cf. A269253, A269254, A298675, A298677, A298678, A299045, A299071. %K A299102 nonn,hard,more,new %O A299102 1,2 %A A299102 _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Feb 02 2018 %I A299101 %S A299101 2,3,5,6,8,9,15,18,23,53,114,194,564,575,585,2594,3143,4578,4970,9261, %T A299101 11508,13298,30018,54993 %N A299101 Indices k such that s_k(5) is a (probable) prime, where s_k(5) = 5*s_{k-1}(5) - s_{k-2}(5), k >= 2, s_0(5) = 1, s_1(5) = 6. %C A299101 From a problem in A269254. %Y A299101 Cf. A269254, A030221, A294099 (row 5). %Y A299101 Cf. A285992, A299107, A299109, A299110, A117522, A299100-A299102. %Y A299101 Cf. A269253, A298675, A298677, A298678, A299045, A299071. %K A299101 nonn,hard,more,new %O A299101 1,1 %A A299101 _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Feb 02 2018 %I A299100 %S A299100 1,2,3,6,9,14,18,146,216,293,704,1143,1530,1593,2924,7163,9176,9489, %T A299100 11531,39543,50423,60720,62868 %N A299100 Indices k such that s_k(4) is a (probable) prime, where s_k(4) = 4*s_{k-1}(4) - s_{k-2}(4), k >= 2, s_0(4) = 1, s_1(4) = 5. %C A299100 From a problem in A269254. %t A299100 s[k_, m_] := s[k, m] = Which[k == 0, 1, k == 1, 1 + m, True, m s[k - 1, m] - s[k - 2, m]]; Select[Range@ 2000, PrimeQ@ Abs@ s[#, 4] &] (* _Michael De Vlieger_, Feb 03 2018 *) %Y A299100 Cf. A269254, A001834, A294099 (row 4). %Y A299100 Cf. A285992, A299107, A299109, A299110, A117522, A299100-A299102. %Y A299100 Cf. A269253, A298675, A298677, A298678, A299045, A299071. %K A299100 nonn,hard,more,new %O A299100 1,2 %A A299100 _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Feb 02 2018 %I A299071 %S A299071 18,52,110,123,198,488,702,724,843,970,1298,1692,2158,2525,3330,4048, %T A299071 4862,5778,6726,6802,7940,9198,10084,10582,13752,15550,17498,19602, %U A299071 21868,24302,26910,29698,30248,32672,35838,39603,42770,46548,50542 %N A299071 Union_{odd primes p, n >= 3} {T_p(n)}, where T_m(x) = x*T_{m-1}(x) - T_{m-2}(x), m >= 2, T_0(x) = 2, T_1(x) = x (dilated Chebyshev polynomials of the first kind). %C A299071 From a problem in A269254. %C A299071 Sequence avoids numbers of the form T_p(T_2(j)). %Y A299071 Cf. A008865 (T_2(n)), A298878 (T_p(n), p prime). %Y A299071 Cf. A285992, A299107, A299109, A299110, A117522, A299100--A299102. %Y A299071 Cf. A269253, A269254, A294099, A298675, A298677, A298678, A299045, A299071. %K A299071 nonn,new %O A299071 1,1 %A A299071 _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Feb 01 2018 %I A299045 %S A299045 1,1,0,1,-1,-1,1,-2,1,1,1,-3,5,-1,0,1,-4,11,-13,1,-1,1,-5,19,-41,34, %T A299045 -1,1,1,-6,29,-91,153,-89,1,0,1,-7,41,-169,436,-571,233,-1,-1,1,-8,55, %U A299045 -281,985,-2089,2131,-610,1,1,1,-9,71,-433,1926,-5741,10009,-7953,1597,-1,0 %N A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor((3*j + k)/2)*binomial(floor((k + j)/2), j)*(-n)^j, n >= 1, k >= 0, read by antidiagonals. %C A299045 This array is used to compute A269252: A269252(n) = least k such that |A(n,k)| is a prime, or -1 if no such k exists. %F A299045 G.f. for row n: (1 + x)/(1 + n*x + x^2), n >= 1. %e A299045 Array begins: %e A299045 1 0 -1 1 0 -1 1 0 -1 1 %e A299045 1 -1 1 -1 1 -1 1 -1 1 -1 %e A299045 1 -2 5 -13 34 -89 233 -610 1597 -4181 %e A299045 1 -3 11 -41 153 -571 2131 -7953 29681 -110771 %e A299045 1 -4 19 -91 436 -2089 10009 -47956 229771 -1100899 %e A299045 1 -5 29 -169 985 -5741 33461 -195025 1136689 -6625109 %e A299045 1 -6 41 -281 1926 -13201 90481 -620166 4250681 -29134601 %e A299045 1 -7 55 -433 3409 -26839 211303 -1663585 13097377 -103115431 %e A299045 1 -8 71 -631 5608 -49841 442961 -3936808 34988311 -310957991 %e A299045 1 -9 89 -881 8721 -86329 854569 -8459361 83739041 -828931049 %t A299045 (* Array: *) %t A299045 Grid[Table[LinearRecurrence[{-n, -1}, {1, 1 - n}, 10], {n, 10}]] %t A299045 (*Array antidiagonals flattened (gives this sequence):*) %t A299045 A299045[n_, k_] := Sum[(-1)^Floor[(3 j + k)/2] Binomial[Floor[(k + j)/2], j] (-n)^j, {j, 0, k}];Flatten[Table[A299045[n - k, k], {n, 11}, {k, 0, n - 1}]] %Y A299045 Cf. A285992, A299107, A299109, A299110, A117522, A299100, A299101, A299102. %Y A299045 Cf. A269251, A269252, A269253, A269254, A294099, A298675, A298677, A298678, A299045, A299071. %Y A299045 Cf. A094954 (unsigned version of this array, but missing the first row). %Y A299045 Cf. Rows: A057078, A033999, A099496, A079935 (or A001835), A004253, A001653, A049685, A070997, A070998, A138288 (or A072256), ... %Y A299045 Cf. Columns: A000012, A001477 (A000027), A110331 (A165900), A123972, A192398, ... %K A299045 sign,tabl,new %O A299045 1,8 %A A299045 _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Feb 01 2018 %I A298878 %S A298878 -2,-1,0,1,2,7,14,18,23,34,47,52,62,79,98,110,119,123,142,167,194,198, %T A298878 223,254,287,322,359,398,439,482,488,527,574,623,674,702,724,727,782, %U A298878 839,843,898,959,970,1022,1087,1154,1223,1294,1298,1367,1442,1519,1598 %N A298878 Union_{p prime, n >= 0} {T_p(n)}, where T_m(x) = x*T_{m-1}(x) - T_{m-2}(x), m >= 2, T_0(x) = 2, T_1(x) = x (dilated Chebyshev polynomials of the first kind). %C A298878 From a problem in A269254. %Y A298878 Cf. A008865 (T_2(n)), A299071. %Y A298878 Cf. A285992, A299107, A299109, A299110, A117522, A299100, A299101, A299102. %Y A298878 Cf. A269253, A269254, A294099, A298675, A298677, A298678, A299045, A299071. %K A298878 sign,new %O A298878 1,1 %A A298878 _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Jan 27 2018 %I A297742 %S A297742 0,1,-1,-1,0,1,1,1,-1,-1,0,-1,-1,1,1,0,1,2,0,-2,-1,0,1,1,-2,-2,1,1,0, %T A297742 -1,-2,1,4,1,-2,-1,0,0,1,2,-1,-4,-1,2,1,0,0,0,-1,-2,1,4,1,-2,-1,0,0,0, %U A297742 -1,-1,3,3,-3,-3,1,1,0,0,0,1,2,-2,-6,0,6,2,-2,-1 %N A297742 Coefficients of polynomial whose zeros are the Möbius function. %C A297742 Also determinant polynomial whose roots are the Möbius function A008683, see formula section. %C A297742 The table (A054431 - x*A051731) starts: %C A297742 { %C A297742 {1 - x, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, %C A297742 {1 - x, -x, 1, 0, 1, 0, 1, 0, 1, 0, 1}, %C A297742 {1 - x, 1, -x, 1, 1, 0, 1, 1, 0, 1, 1}, %C A297742 {1 - x, -x, 1, -x, 1, 0, 1, 0, 1, 0, 1}, %C A297742 {1 - x, 1, 1, 1, -x, 1, 1, 1, 1, 0, 1}, %C A297742 {1 - x, -x, -x, 0, 1, -x, 1, 0, 0, 0, 1}, %C A297742 {1 - x, 1, 1, 1, 1, 1, -x, 1, 1, 1, 1}, %C A297742 {1 - x, -x, 1, -x, 1, 0, 1, -x, 1, 0, 1}, %C A297742 {1 - x, 1, -x, 1, 1, 0, 1, 1, -x, 1, 1}, %C A297742 {1 - x, -x, 1, 0, -x, 0, 1, 0, 1, -x, 1}, %C A297742 {1 - x, 1, 1, 1, 1, 1, 1, 1, 1, 1, -x} %C A297742 } %F A297742 Let A be the lower triangular matrix: if n mod k = 0 then 1 else 0. %F A297742 Let B the upper triangular matrix: if k mod n = 0 then A008683(n) else 0. %F A297742 The polynomial is then: determinant(A.B - x*A) where . stands for matrix multiplication and * stands for normal multiplication like 2*3=6. x is the variable to solve for: polynomial = determinant(A054431 - x*A051731). %e A297742 The table of polynomial coefficients starts: %e A297742 { %e A297742 { 0}, %e A297742 { 1, -1}, %e A297742 {-1, 0, 1}, %e A297742 { 1, 1, -1, -1}, %e A297742 { 0, -1, -1, 1, 1}, %e A297742 { 0, 1, 2, 0, -2, -1}, %e A297742 { 0, 1, 1, -2, -2, 1, 1}, %e A297742 { 0, -1, -2, 1, 4, 1, -2, -1}, %e A297742 { 0, 0, 1, 2, -1, -4, -1, 2, 1}, %e A297742 { 0, 0, 0, -1, -2, 1, 4, 1, -2, -1}, %e A297742 { 0, 0, 0, -1, -1, 3, 3, -3, -3, 1, 1}, %e A297742 { 0, 0, 0, 1, 2, -2, -6, 0, 6, 2, -2, -1} %e A297742 } %t A297742 (* program 1 *) %t A297742 Clear[x, P] %t A297742 TableForm[polynomial = Table[ %t A297742 A = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}]; %t A297742 B = Table[ %t A297742 Table[If[Mod[k, n] == 0, MoebiusMu[n], 0], {k, 1, nn}], {n, 1, %t A297742 nn}]; %t A297742 Det[A.B - x*A], {nn, 1, 11}]]; %t A297742 Flatten[CoefficientList[polynomial, x]] %Y A297742 Cf. A008683, A051731, A054431. %K A297742 sign,tabl,new %O A297742 0,18 %A A297742 _Mats Granvik_, Jan 05 2018 %I A297477 %S A297477 0,1,-1,-2,0,1,3,3,-1,-1,-4,-8,-3,2,1,5,15,14,2,-3,-1,-6,-24,-35,-20, %T A297477 0,4,1,7,35,69,65,25,-3,-5,-1,-8,-48,-119,-154,-105,-28,7,6,1,9,63, %U A297477 188,308,294,154,28,-12,-7,-1,-10,-80,-279,-552,-672,-504,-210,-24,18,8,1 %N A297477 Triangle read by rows: T(n, k) gives the coefficients of x^k of the characteristic polynomial P(n, x) of the n X n matrix M with entries M(i, j) = 1 if i = 1 or j = 1, -1 if i = j > 1, and 0 otherwise. T(0, 0):= 0. %C A297477 The norm of the matrix M appears to be sqrt(n), where with norm is meant the eigenvalue of the largest magnitude, negative or positive. Row sums appear to be A085750 [see below for the proof]. %C A297477 Also the coefficients of the characteristic polynomial of the matrix defined by the recurrence: A(n, k) = if n < k then if and(n > 1, k > 1) then Sum_{i=1..k-1} -A(k-i, n) else 0 else if and(n > 1, k > 1) then Sum_{i=1..n-1} -A(n-i, k) else 0. %C A297477 By letting the upper summation indexes "k-1" and "n-1" in the recurrence above, change place with each other one gets the number theoretic matrix A191898, and it appears that the eigenvalue norm sqrt(n) of this matrix is a lower bound for the eigenvalue norm of matrix A191898 which in turn for n>10 appears to be close to A007917, the previous prime sequence. If the eigenvalue norm of matrix A191898 also can be proven to be less than n+1, then one could say that there is always a prime gap between sqrt(n) and n+1. %C A297477 From _Wolfdieter Lang_, Feb 32 2018: (Start) %C A297477 The characteristic polynomial P(n, x) = Det(M_n - x*1_n), with the n X n matrix M_n defined in the name and 1_n the n dimensional unit matrix, satisfies, after expanding the last row, the recurrence: P(n, x) = -z*P(n-1, x) + (-1)^(n-1)*z^(n-2), for n >= 2, and input P(1, x) = y, where y = 1-x and z = 1+x. The solution is P(n, x) = y*(-z)^(n-1) - (n-1)*(-z)^(n-2) = (-1)^n*(1 + x)^(n-2)*(x^2 - n), for n >= 1. After picking the coefficient of x^k this becomes the formula for T(n, k) given in the formula section. %C A297477 The Determinant of M_n is P(n, 0) = T(n, 0) = (-1)^n*n = A181983(n). %C A297477 The eigenvalues of M_n are +1 for n = 1 and for n >= 2 they are +sqrt(n), -sqrt(n), and n-2 times -1. %C A297477 Therefore the spectral radius (absolute value of the maximal eigenvalue) is rho_n = sqrt(n), and the spectral norm of M_n (square root of the maximal eigenvalue of (M_n)^+ M_n is also sqrt(n), for n >= 1. See the conjecture in the first comment above. %C A297477 The square of the Frobenius norm (aka Hilbert-Schmidt norm) of M_n is max_{i,j=1..n} |M_n(i,j)|^2 = 3*n - 2 = A016777(n-1), for n >= 1. %C A297477 The row sums are P(n, 1) = (-1)^(n-1)*(n-1)*2^(n-2) = A085750(n), for n >= 1, and for n=0 the row sum is 0. The alternating row sums are P(n, -1) = 2 for n=1, -1 for n = 2, and zero otherwise. %C A297477 The column sequence (without leading zero) for k = 1 is (-1)^(n+1)*n*(n-2), for n >= 1, which is -A131386(n). For k = 2 it is (-1)^n*(1 - n*binomial(n-2, 2)) for n >= 2 which is (-1)^n*A110427(n-1). Other columns follow from the formula for T(n, k). (End) %F A297477 From _Wolfdieter Lang_, Feb 02 2018: (Start) %F A297477 T(n, k) = [x*k] P(n, x), for n >= 1, with P(n, x) = Det(M_n - x*1_n), and the matrix M_n defined in the name (1_n is the n dimensional unit matrix). T(0, 0):= 0. %F A297477 T(n, k) = (-1)^(n+1)*n for k = 0, (-1)^(n+1)*n*(n-2) for k = 1, and (-1)^n*(binomial(n-2, k-2) - n*binomial(n-2, k)) for k >= 2, with n >= 0 and 0 <= k <= n. T(n, k) = 0 for k > n. (End) %e A297477 The matrix for these characteristic polynomials starts: %e A297477 { %e A297477 {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, %e A297477 {1, -1, 0, 0, 0, 0, 0, 0, 0, 0}, %e A297477 {1, 0, -1, 0, 0, 0, 0, 0, 0, 0}, %e A297477 {1, 0, 0, -1, 0, 0, 0, 0, 0, 0}, %e A297477 {1, 0, 0, 0, -1, 0, 0, 0, 0, 0}, %e A297477 {1, 0, 0, 0, 0, -1, 0, 0, 0, 0}, %e A297477 {1, 0, 0, 0, 0, 0, -1, 0, 0, 0}, %e A297477 {1, 0, 0, 0, 0, 0, 0, -1, 0, 0}, %e A297477 {1, 0, 0, 0, 0, 0, 0, 0, -1, 0}, %e A297477 {1, 0, 0, 0, 0, 0, 0, 0, 0, -1} %e A297477 } %e A297477 ---------------------------------------------------------------------- %e A297477 The table T(n, k) begins: %e A297477 n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 .. %e A297477 0: 0 %e A297477 1: 1 -1 %e A297477 2: -2 0 1 %e A297477 3: 3 3 -1 -1 %e A297477 4: -4 -8 -3 2 1 %e A297477 5; 5 15 14 2 -3 -1 %e A297477 6: -6 -24 -35 -20 0 4 1 %e A297477 7: 7 35 69 65 25 -3 -5 -1 %e A297477 8: -8 -48 -119 -154 -105 -28 7 6 1 %e A297477 9: 9 63 188 308 294 154 28 -12 -7 -1 %e A297477 10: -10 -80 -279 -552 -672 -504 -210 -24 18 8 1 %e A297477 11: 11 99 395 915 1350 1302 798 270 15 -25 -9 -1 %e A297477 12: -12 -120 -539 -1430 -2475 -2904 -2310 -1188 -330 0 33 10 1 %e A297477 ... reformatted by _Wolfdieter Lang_, Feb 02 2018. %p A297477 f:= proc(n) local M,P,lambda,k; %p A297477 M:= Matrix(n,n, proc(i,j) if i=1 or j=1 then 1 elif i=j then -1 else 0 fi end proc); %p A297477 P:= (-1)^n*LinearAlgebra:-CharacteristicPolynomial(M,lambda); %p A297477 seq(coeff(P,lambda,k),k=0..n) %p A297477 end proc: %p A297477 f(0):= 0: %p A297477 for n from 0 to 10 do f(n) od; # _Robert Israel_, Feb 02 2018 %t A297477 Clear[A, x, t]; %t A297477 Table[t[n_, 1] = 1; %t A297477 t[1, k_] = 1; %t A297477 t[n_, k_] := %t A297477 t[n, k] = %t A297477 If[n < k, %t A297477 If[And[n > 1, k > 1], Sum[-t[k - i, n], {i, 1, k - 1}], 0], %t A297477 If[And[n > 1, k > 1], Sum[-t[n - i, k], {i, 1, n - 1}], 0]]; %t A297477 A = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]; %t A297477 CoefficientList[CharacteristicPolynomial[A, x], x], {nn, 1, 10}]; %t A297477 Flatten[%] %Y A297477 Cf. A016777, A085750 (row sums), A067998, A110427 (column k=2), -A131386 (column k=1), A181983 (Det M_n), A191898. %K A297477 tabl,sign,easy,new %O A297477 0,4 %A A297477 _Mats Granvik_, Dec 30 2017 %E A297477 Edited by _Wolfdieter Lang_, Feb 02 2018. %I A299068 %S A299068 3,4,8,7,11,6,10,12,11,9,9,9,13,22,12,7,7,11,21,28,9,7,17,14,13,14,13, %T A299068 13,11,9,10,12,17,33,28,8,7,20,19,15,9,10,21,29,10,7,14,19,18,21,11,9, %U A299068 16,44,46,14,7,9,15,9,9,18,40,24,18,8,9,30,18,17,11 %N A299068 Number of pairs of factors of n^2*(n^2-1) which differ by n. %C A299068 The question arose when seeking triples of numbers for which the sum of the squares of any two is congruent to 1 modulo the third. %C A299068 From _Robert Israel_, Feb 04 2018: (Start) %C A299068 For n > 7, a(n)>= 7, as there are at least the following pairs: %C A299068 (1,n+1), (n,2*n), (2*n,3*n), ((n^2-n)/2,(n^2+n)/2), (n^2-n,n^2), (n^2,n^2+n), and (3*n, 4*n) (if n is odd) or (n/2,3*n/2) (if n is even). %C A299068 If k in A299159 is sufficiently large, then a(12*k-2)=7. Dickson's conjecture implies there are infinitely many such k, and thus infinitely many n with a(n)=7. (End) %H A299068 Robert Israel, Table of n, a(n) for n = 2..10000 %p A299068 a:= n-> (s-> add(`if`(i+n in s, 1, 0), i=s))( %p A299068 numtheory[divisors](n^2*(n^2-1))): %p A299068 seq(a(n), n=2..100); # _Alois P. Heinz_, Feb 01 2018 %t A299068 Array[With[{d = Divisors[# (# - 1)] &[#^2]}, Count[d + #, _?(MemberQ[d, #] &)]] &, 71, 2] (* _Michael De Vlieger_, Feb 01 2018 *) %Y A299068 Cf. A299159. %K A299068 nonn,new %O A299068 2,1 %A A299068 _John H Mason_, Feb 01 2018 %I A294674 %S A294674 1,3,5,7,11,13,15,17,19,23,29,31,35,37,41,43,47,53,59,61,67,71,73,77, %T A294674 79,83,89,97,101,103,105,107,109,113,127,131,137,139,143,149,151,157, %U A294674 163,167,173,179,181,191,193,197,199,211,221,223,227,229,233,239,241,251,257,263,269,271 %N A294674 Numbers that are the product of any number of consecutive odd primes. %C A294674 If a(n) is an odd squarefree number with no gaps in its prime >= A065091(1) factors, b(n) is an odd squarefree number with no gaps in its prime >= A065091(2) factors, and c(n) is an odd squarefree number with no gaps in its prime >= A065091(3) factors, ..., then a(n) >= b(n) >= c(n) >= ... >= A056911(n). %e A294674 105 is in this sequence because 105 = 3*5*7 = A065091(1)*A065091(2)*A065091(3), where A065091() are odd primes. %t A294674 {1}~Join~Select[Range[3, 275, 2], And[SquareFreeQ@ #, MemberQ[{{}, {1}}, Union@ Differences@ PrimePi@ FactorInteger[#][[All, 1]]]] &] (* _Michael De Vlieger_, Nov 15 2017 *) %o A294674 (PARI) isok(n) = {if ((n % 2) && issquarefree(n), f = factor(n); v = vector(#f~, k, primepi(f[k,1])); for (k=2, #v, if (v[k] - v[k-1] != 1, return (0))); return (1);); return (0);} \\ _Michel Marcus_, Nov 08 2017 %Y A294674 Intersection of A056911 and A073485. %Y A294674 Cf. A065091, A294472. %K A294674 nonn,easy,new %O A294674 1,2 %A A294674 _Juri-Stepan Gerasimov_, Nov 06 2017 %E A294674 a(57) corrected by _Rémy Sigrist_, Nov 18 2017 %I A294472 %S A294472 1,2,3,5,6,7,10,11,13,14,15,17,19,22,23,26,29,30,31,34,35,37,38,41,43, %T A294472 46,47,53,58,59,61,62,67,70,71,73,74,77,79,82,83,86,89,94,97,101,103, %U A294472 105,106,107,109,113,118,122,127,131,134,137,139,142,143,146,149,151,154,157,158,163,166 %N A294472 Squarefree numbers such that all odd prime factors are consecutive primes. %C A294472 The union of products of any number of consecutive odd primes and twice products of any number of consecutive odd primes. %C A294472 If A073485(n) is squarefree numbers with no gaps in their prime >= A000040(1) factors, a(n) is squarefree numbers with no gaps in their prime >= A000040(2) factors, b(n) is squarefree numbers with no gaps in their prime >= A000040(3) factors, ..., x(n) is squarefree numbers with no gaps in their prime >= A000040(y) factors, ..., then also A073485(n) >= a(n) >= b(n) >= ... >= x(n) >= ... >= A005117(n). %C A294472 Conjecture: if z(n) is smallest y such that n*k - k^2 is squarefree number with no gaps in their prime >= A000040(y) factors for some k < n, then z(n) >= 1 for all n > 1. %C A294472 If a(n-1) + 1 = a(n) = a(n+1) - 1, then a(n) equal to 2, 6, 30, 106, etc. %C A294472 Squarefree numbers where any two neighboring odd prime factors in the ordered list of prime factors are consecutive primes. - _Felix Fröhlich_, Nov 01 2017 %e A294472 70 is in this sequence because 2*5*7 = 70 is a squarefree number with two consecutive odd prime factors 5 and 7. %p A294472 N:= 1000: # to get all terms <= N %p A294472 R:= 1,2: %p A294472 Oddprimes:= select(isprime, [seq(i,i=3..N,2)]): %p A294472 for i from 1 to nops(Oddprimes) do %p A294472 p:= 1: %p A294472 for k from i to nops(Oddprimes) do %p A294472 p:= p*Oddprimes[k]; %p A294472 if p > N then break fi; %p A294472 if 2*p <= N then R:= R, p, 2*p %p A294472 else R:= R,p %p A294472 fi %p A294472 od; %p A294472 od: %p A294472 R:= sort([R]); # _Robert Israel_, Nov 01 2017 %t A294472 Select[Range@ 166, And[Union@ #2 == {1}, Or[# == {1}, # == {}] &@ Union@ Differences@ PrimePi@ DeleteCases[#1, 2]] & @@ Transpose@ FactorInteger[#] &] (* _Michael De Vlieger_, Nov 01 2017 *) %Y A294472 Cf. A000040, A005117, A073485, A294674. %K A294472 nonn,new %O A294472 1,2 %A A294472 _Juri-Stepan Gerasimov_, Oct 31 2017 %E A294472 Definition corrected by _Michel Marcus_, Nov 01 2017 %I A293575 %S A293575 -1,0,0,0,0,2,0,1,0,2,0,3,0,2,2,1,0,3,0,3,2,2,0,5,0,2,1,3,0,6,0,2,2,2, %T A293575 2,4,0,2,2,5,0,6,0,3,3,2,0,6,0,3,2,3,0,5,2,5,2,2,0,9,0,2,3,2,2,6,0,3, %U A293575 2,6,0,7,0,2,3,3,2,6,0,6,1,2,0,9,2,2,2,5,0,9,2,3,2,2,2,8,0,3,3,4,0,6,0,5,6 %N A293575 Difference between the number of proper divisors of n and the number of squares dividing n. %C A293575 The difference between the number of ways of writing n = m + k and the number of ways of writing n = r*s, where m|k and r|s. %C A293575 First occurrence of k beginning with k=-1: 1, 2, 8, 6, 12, 36, 24, 30, 72, 96, 60, 2097152, 216, 576, 120, 210, 1152, 240, 864, etc. - _Robert G. Wilson v_, Nov 28 2017 %H A293575 Antti Karttunen, Table of n, a(n) for n = 1..10000 %F A293575 a(n) = A032741(n) - A046951(n). %F A293575 a(n) = A056595(n) - 1. - _Antti Karttunen_, Oct 30 2017 %F A293575 a(n) = 0 iff n is a prime or a square of a prime, A000430. - _Robert G. Wilson v_, Nov 28 2017 %e A293575 a(6) = 2 because 2 is difference of number of ways of writing n = 1 + 5 = 2 + 4 = 3 + 3 where 1|5, 2|4, 3|3 and number of ways of writing n = 1*6 where 1|6. %t A293575 f[n_] := Block[{d = Divisors@ n}, Length@ d - Length[ Select[ d, IntegerQ@ Sqrt@# &]] - 1];; Array[f, 105] (* _Robert G. Wilson v_, Nov 28 2017 *) %Y A293575 One less than A056595. %Y A293575 Cf. A000430, A032741, A046951, A166684, A080257. %K A293575 sign,new %O A293575 1,6 %A A293575 _Juri-Stepan Gerasimov_, Oct 14 2017 %I A297616 %S A297616 1,2,3,3,4,3,4,4,4,3,4,4,5,4,4,4,5,5,6,6,6,5,6,6,6,5,5,5,6,6,7,7,7,6, %T A297616 6,6,7,6,6,6,7,7,8,8,8,7,8,8,8,8,8,8,9,9,9,9,9,8,9,9,10,9,9,9,9,9,10, %U A297616 10,10,10,11,11,12,11,11,11,11,11,12,12,12,11,12,12,12,11,11,11,12,12,12,12,12,11,11,11,12 %N A297616 a(n) is the number of connected components in the graph with vertices 1..n and adjacency criterion i and j not coprime. %F A297616 a(n) = 1 + pi(n) - pi(n / 2) + [n >= 4], where pi denotes the prime counting function (A000720, generalized to reals), and [] the Iverson bracket. %t A297616 A[n_] := Length[ %t A297616 ConnectedComponents[ %t A297616 AdjacencyGraph[Map[Boole[# != 1] &, Array[GCD, {n, n}], {2}]]]] %t A297616 Table[A[n], {n, 1, 107}] %o A297616 (PARI) a(n) = 1 + primepi(n) - primepi(n / 2) + (n >= 4); \\ _Michel Marcus_, Jan 09 2018 %Y A297616 Cf. A000720. %K A297616 nonn,new %O A297616 1,2 %A A297616 _Luc Rousseau_, Jan 01 2018 %I A298950 %S A298950 1,4,8,17,25,40,52,73,89,116,136,169,193,232,260,305,337,388,424,481, %T A298950 521,584,628,697,745,820,872,953,1009,1096,1156,1249,1313,1412,1480, %U A298950 1585,1657,1768,1844,1961,2041,2164,2248,2377,2465,2600,2692,2833,2929,3076,3176,3329,3433 %N A298950 Numbers k such that 5*k - 4 is a square. %C A298950 a(n) is a member of A140612. Proof: a(n) = n^2 + (n/2-1)^2 for even n, otherwise a(n) = (n-1)^2 + ((n+1)/2)^2; also, a(n) + 1 = (n-1)^2 + (n/2+1)^2 for even n, otherwise a(n) + 1 = n^2 + ((n-3)/2)^2. Therefore, both a(n) and a(n) + 1 belong to A001481. %C A298950 Primes in sequence are listed in A245042. %C A298950 Squares in sequence are listed in A081068. %H A298950 Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1). %F A298950 G.f.: x*(1 + x^2)*(1 + 3*x + x^2)/((1 - x)^3*(1 + x)^2). %F A298950 a(n) = a(1-n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). %F A298950 a(n) = (10*n*(n-1) + (2*n-1)*(-1)^n + 9)/8. %F A298950 a(n) = A036666(n) + 1. %t A298950 Table[(10 n (n - 1) + (2 n - 1) (-1)^n + 9)/8, {n, 1, 60}] %o A298950 (PARI) Vec((1+x^2)*(1+3*x+x^2)/((1-x)^3*(1+x)^2)+O(x^60)) %o A298950 (PARI) vector(60, n, nn; (10*n*(n-1)+(2*n-1)*(-1)^n+9)/8) %o A298950 (Sage) [(10*n*(n-1)+(2*n-1)*(-1)^n+9)/8 for n in (1..60)] %o A298950 (Maxima) makelist((10*n*(n-1)+(2*n-1)*(-1)^n+9)/8, n, 1, 60); %o A298950 (MAGMA) [(10*n*(n-1)+(2*n-1)*(-1)^n+9)/8: n in [1..60]]; %Y A298950 Cf. A195162: numbers k such that 5*k + 4 is a square. %Y A298950 Subsequence of A001481, A020668, A036404, A140612. %Y A298950 Cf. A036666, A081068, A106833 (first differences), A245042. %K A298950 nonn,easy,new %O A298950 1,2 %A A298950 _Bruno Berselli_, Jan 30 2018 %I A296105 %S A296105 1,2,5,25,157,1325,14358,199763,3549001,80673244,2352747542, %T A296105 88240542454,4261209044877,264988507673267,21207485269909946, %U A296105 2182146922863398203 %N A296105 a(n) is the number of connected transitive relations over n unlabeled nodes. %C A296105 Inverse Euler transform of A091073. Here "connected" means that it is possible to reach any vertex starting from any other vertex by traversing edges in some direction, i.e., not necessarily in the direction in which the edges point, as in weakly connected digraphs. %e A296105 a(2) = 5 because there are five connected transitive relations up to isomorphism: a->b with no loops, a->b with a loop on a, a->b with a loop on b, a->b->a with no loops, and a->b->a with loops on both a and b. %Y A296105 Cf. A091073 (all unlabeled transitive relations). For the labeled case, see A245731 (connected labeled transitive relations) and A006905 (all labeled transitive relations). %K A296105 nonn,more,new %O A296105 0,2 %A A296105 _Daniele P. Morelli_, Dec 04 2017 %I A294148 %S A294148 2,3,7,19,83,757,20849,3010457,5223344167,377505222176491, %T A294148 7334735304307198091659,628169243329433959747511898881729, %U A294148 15743997159315671181189678367886578609038417676391,62470143575895304690094463208359121879929322420894118855801094477279900397 %N A294148 a(n) is the smallest prime whose square is greater than the cube of a(n-1); a(1) = 2. %C A294148 a(n) is the smallest prime such that a(n) > a(n-1)^(3/2); a(1) = 2. %H A294148 Robert G. Wilson v, Table of n, a(n) for n = 1..20 %e A294148 a(2) = 3 because 3^2 = 9 > a(1)^3 = 8 and 3 is the smallest such prime. %e A294148 a(3) = 7 because 7^2 = 49 > a(2)^3 = 27 and 7 is the smallest such prime. %t A294148 f[s_List] := Append[s, NextPrime[ Sqrt[s[[-1]]^3]]]; s = {2}; Nest[f, s, 13] (* _Robert G. Wilson v_, Nov 18 2017 *) %o A294148 (PARI) { %o A294148 p=2;print1(p", "); %o A294148 for(n=1,10, %o A294148 p1=nextprime(p+1); %o A294148 while(p^3>p1^2,p1=nextprime(p1+1)); %o A294148 p=p1;print1(p1", ") %o A294148 ) %o A294148 } %o A294148 (PARI) lista(nn) = {a=2; print1(a, ", "); for (n=1, nn, a=nextprime(sqrtint(a^3)+1); print1(a, ", "));} \\ _Michel Marcus_, Oct 29 2017 %Y A294148 Cf. A001248 and A030078 (squares and cubes of primes). %Y A294148 Cf. A059842 (similar with n instead of prime(n)). %K A294148 nonn,new %O A294148 1,1 %A A294148 _Dimitris Valianatos_, Oct 23 2017 %E A294148 a(10)-a(14) from _Michel Marcus_, Oct 29 2017 %I A294064 %S A294064 5,7,13,35,43,55,77,127,133,155,167,253,287,295,365,475,497,533,595, %T A294064 713,1007,1177,1483,1805,2323,2575,2723,2927,3107,3415,3487,3823,4145, %U A294064 4213,4367,4565,4717,4927,4963,5125,5215,5363,5417,5587,5627,5795,6133,6587,6797 %N A294064 Numbers n such that 2*n - 3, 2*n + 3, 3*n - 2, 3*n + 2 are primes. %C A294064 The common numbers of A098090, A067076, A153183, A024893. %C A294064 Conjecture: The Sum_{n>=1} 1/a(n) = 0.57... converges. %C A294064 Note that the sum of the 4 primes that are obtained is 10 times the original term: (2*n - 3) + (2*n + 3) + (3*n - 2) + (3*n + 2) = 10*n. %C A294064 From _Robert G. Wilson v_, Nov 19 2017: (Start) %C A294064 Number of terms less than 10^k: 2, 7, 20, 55, 189, 919, 4863, 28218, 174469, ..., ; %C A294064 Number of prime terms less than 10^k: 2, 4, 6, 12, 39, 140, 558, 2755, 14804, ..., . %C A294064 All terms are == {5, 7, 13, 17, 23, 25} (mod 30). %C A294064 (End) %H A294064 Robert G. Wilson v, Table of n, a(n) for n = 1..10000 %e A294064 5 is in the sequence because 2*5-3 = 7, 2*5+3 = 13, 3*5-2 = 13, 3*5+2 = 17 and the tetrad [7, 13, 13, 17] are all prime numbers. %e A294064 7 is in the sequence because 2*7-3 = 11, 2*7+3 = 17, 3*7-2 = 19, 3*7+2 = 23 and the tetrad [11, 17, 19, 23] are all prime numbers. %p A294064 P:=proc(n) if isprime(2*n-3) and isprime(2*n+3) and isprime(3*n-2) and isprime(3*n+2) %p A294064 then n; fi; end: seq(P(i),i=1..6000); # _Paolo P. Lava_, Oct 31 2017 %t A294064 Select[Range[10^4], Function[k, AllTrue[Flatten@ Map[#1 k + {-1, 1} #2 & @@ # &, {#, Reverse@ #}] &@ {2, 3}, PrimeQ]]] (* _Michael De Vlieger_, Oct 22 2017 *) %o A294064 (PARI) { %o A294064 for(n=1,10000, %o A294064 if(isprime(2*n-3)&&isprime(2*n+3)&&isprime(3*n-2)&&isprime(3*n+2), %o A294064 print1(n", ") %o A294064 ) %o A294064 ) %o A294064 } %Y A294064 Cf. A098090, A067076, A153183, A024893. %K A294064 nonn,new %O A294064 1,1 %A A294064 _Dimitris Valianatos_, Oct 22 2017 %I A293990 %S A293990 1,3,3,5,7,9,9,11,13,15,15,17,19,21,21,23,25,27,27,29,31,33,33,35,37, %T A293990 39,39,41,43,45,45,47,49,51,51,53,55,57,57,59,61,63,63,65,67,69,69,71, %U A293990 73,75,75,77,79,81,81,83,85,87,87,89,91,93,93 %N A293990 a(n) = (3*n + ((n-2) mod 4))/2. %C A293990 The product (2/3) * (4/3) * (6/5) * (6/7) * (8/9) * (10/9) * (12/11) * (12/13) * ... = Pi/(2*sqrt(3)). The denominators are a(n) for n >= 1 and numerators are a(n-1) + A093148(n) for n >= 1 -> [2, 4, 6, 6, 8, 10, 12, 12, ...]. %C A293990 Let r(n) = (a(n)-1)/(a(n)+1)) if a(n) mod 4 = 1, (a(n)+1)/(a(n)-1)) otherwise; then Product_{n>=1} r(n) = (2/1) * (2/1) * (2/3) * (4/3) * (4/5) * (4/5) * (6/5) * (6/7) * ... = Pi*sqrt(3)/2 = 2.72069904635132677... %C A293990 The odd numbers of partial sums this sequence, are identified with the A003215 sequence. Also the prime numbers that appear in partial sums in this sequence, are identified with the A002407 sequence. %H A293990 Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1). %F A293990 Sum_{n>=0} 1/a(n)^2 = 5*Pi^2/36 = 1.3707783890401886970... = 10*A086729. %F A293990 (a(n) - n) * (-1)^(n+1) = A134967(n) for n >= 0. %F A293990 a(n) - n = A162330(n) for n >= 0. %F A293990 a(n) - n = A285869(n+1) for n >= 0. %F A293990 a(n) + a(n+1) = A157932(n+2) for n >= 0. %F A293990 a(n) + (2*n+1) = A047298(n+1) for n >= 0. %F A293990 From _Colin Barker_, Oct 21 2017: (Start) %F A293990 G.f.: x*(1 + 2*x + 2*x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)). %F A293990 a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5. %F A293990 (End) %F A293990 a(n + 8) = a(n) + 12. - _David A. Corneth_, Oct 21 2017 %F A293990 a(4*k+4) * a(4*k+3) - a(4*k+2) * a(4*k+1) = 2*A063305(k+3) for k >= 0. %F A293990 Sum_{n>=0} 1/(a(n) + a(n+2))^2 = (4*Pi^2 - 27) / 108 = (A214549 - 1) / 4. %p A293990 A293990:=n->(3*n+((n-2) mod 4))/2: seq(A293990(n), n=0..100); # _Wesley Ivan Hurt_, Oct 29 2017 %t A293990 Table[(3*n + Mod[(n - 2), 4])/2, {n, 0, 100}] (* _Wesley Ivan Hurt_, Oct 29 2017 *) %t A293990 f[n_] := (3n + Mod[n - 2, 4])/2; Array[f, 65, 0] (* or *) %t A293990 LinearRecurrence[{1, 0, 0, 1, -1}, {1, 3, 3, 5, 7}, 65] (* or *) %t A293990 CoefficientList[ Series[(x^4 + 2x^3 + 2x + 1)/((x - 1)^2 (x^3 + x^2 + x + 1)), {x, 0, 64}], x] (* _Robert G. Wilson v_, Nov 28 2017 *) %o A293990 (PARI) a(n) = (3*n + (n-2)%4) / 2 %o A293990 (PARI) Vec(x*(1 + 2*x + 2*x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^30)) \\ _Colin Barker_, Oct 21 2017 %o A293990 (PARI) first(n) = my(start=[1,3,3,5,7,9,9,11]); if(n<=8, return(start)); my(res=vector(n)); for (i=1, 8, res[i] = start[i]); for(i = 1, n-8 ,res[i+8] = res[i] + 12); res \\ _David A. Corneth_, Oct 21 2017 %o A293990 (MAGMA) [(3*n+((n-2) mod 4))/2 : n in [0..100]]; // _Wesley Ivan Hurt_, Oct 29 2017 %Y A293990 Cf. A007310, A047298, A063305, A093148, A134967, A157932, A162330, A168329, A214549, A285869, A003215, A002407. %K A293990 nonn,easy,new %O A293990 0,2 %A A293990 _Dimitris Valianatos_, Oct 21 2017 %I A295738 %S A295738 2,3,5,7,13,43,61,283,1669,2316667,3670169811199621, %T A295738 21880301185536674566743742843, %U A295738 554620380869291027814931305550350952069,846453153412475180263654973437331373840097042541795707 %N A295738 a(1)=2, a(2)=3; for n > 2, a(n) is the smallest prime number greater than a(n-1) which has the form a(n-2)*2^k - a(n-1) for some integer k. %C A295738 a(15) = 1128606171...0947680901 contains 276 digits. %C A295738 a(16) = 2855795709...8400549243 contains 4622 digits. - _Robert G. Wilson v_, Nov 30 2017 %C A295738 The corresponding k are 2, 2, 2, 3, 3, 3, 5, 13, 41, 73, 77, 85, 785, ... %C A295738 If the condition a(1) < a(2) < a(3) < ... is not satisfied, the sequence becomes 2, 3, 5, 7, 3, 11, 13, 31, 73, 919, ... where a(5) = 3 = 5*2^1 - 7. %C A295738 Conjecture 1: %C A295738 For n > 1, a(2n)== 7 (mod 12) and a(2n+1)== 1 (mod 12); %C A295738 for n > 2, a(2n)== 19 (mod 24) and a(2n-1)== 13 (mod 24). %C A295738 Conjecture 2: %C A295738 For n > 1, L(a(2n)/a(2n+1)) = -1 and L(a(2n+1)/a(2n+2)) = 1 where L(x/y) is the Legendre symbol. %H A295738 Robert G. Wilson v, Table of n, a(n) for n = 1..15 %e A295738 5 = 2*2^2 - 3; 7 = 3*2^2 - 5; 13 = 5*2^2 - 7; 43 = 7*2^3 - 13. %p A295738 p1:=2:p2:=3: %p A295738 for n from 1 to 12 do: %p A295738 ii:=0: %p A295738 for k from 0 to 10^6 while(ii=0) do: %p A295738 p3:=p1*2^k-p2: %p A295738 if p3=floor(p3) and isprime(p3) and p3 > p2 %p A295738 then %p A295738 ii:=1:p1:=p2:p2:=p3:printf(`%d, `,p3): %p A295738 else %p A295738 fi: %p A295738 od: %p A295738 od: %t A295738 f[s_List] := Block[{k = 0, p = s[[-2]], q = s[[-1]]}, While[r = p*2^k - q; r <= q || ! PrimeQ@r, k++]; Append[s, r]]; s = {2, 3}; Nest[f, s, 12] (* _Robert G. Wilson v_, Nov 30 2017 *) %o A295738 (PARI) first(n) = { my(res = vector(n)); res[1]=2; res[2]=3; for(x=3, n, for(k=0, +oo, my(p=res[x-2]*2^k-res[x-1]); if(isprime(p) && p > res[x-1], res[x]=p; break()))); res; } \\ _Iain Fox_, Nov 29 2017 %Y A295738 Cf. A000040. %K A295738 nonn,new %O A295738 1,1 %A A295738 _Michel Lagneau_, Nov 26 2017 %I A299142 %S A299142 0,1,1,1,4,1,2,18,18,2,3,64,129,64,3,5,236,899,899,236,5,8,888,6205, %T A299142 11179,6205,888,8,13,3336,43066,143548,143548,43066,3336,13,21,12512, %U A299142 298361,1850266,3426869,1850266,298361,12512,21,34,46928,2068149 %N A299142 T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299142 Table starts %C A299142 ..0.....1........1..........2.............3...............5.................8 %C A299142 ..1.....4.......18.........64...........236.............888..............3336 %C A299142 ..1....18......129........899..........6205...........43066............298361 %C A299142 ..2....64......899......11179........143548.........1850266..........23808476 %C A299142 ..3...236.....6205.....143548.......3426869........81988764........1958821107 %C A299142 ..5...888....43066....1850266......81988764......3643124959......161617794805 %C A299142 ..8..3336...298361...23808476....1958821107....161617794805....13311860331263 %C A299142 .13.12512..2068149..306389599...46801360032...7170422794173..1096571119921011 %C A299142 .21.46928.14334327.3942948157.1118229413140.318133492126048.90332592148780928 %H A299142 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299142 Empirical for column k: %F A299142 k=1: a(n) = a(n-1) +a(n-2) %F A299142 k=2: a(n) = 4*a(n-1) -2*a(n-2) +4*a(n-3) for n>4 %F A299142 k=3: [order 10] for n>11 %F A299142 k=4: [order 33] for n>34 %e A299142 Some solutions for n=5 k=4 %e A299142 ..0..0..1..0. .0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..0..0 %e A299142 ..1..0..0..1. .0..1..0..0. .1..1..1..0. .0..0..1..0. .0..1..0..1 %e A299142 ..0..1..0..1. .0..1..1..0. .0..0..0..1. .0..0..0..1. .1..0..1..0 %e A299142 ..0..1..0..0. .0..0..1..1. .0..0..1..0. .1..1..0..1. .0..1..0..1 %e A299142 ..0..1..1..1. .0..0..0..1. .1..1..0..0. .1..0..0..1. .1..0..0..0 %Y A299142 Column 1 is A000045(n-1). %Y A299142 Column 2 is A231950(n-1). %K A299142 nonn,tabl,new %O A299142 1,5 %A A299142 _R. H. Hardin_, Feb 03 2018 %I A299141 %S A299141 8,3336,298361,23808476,1958821107,161617794805,13311860331263, %T A299141 1096571119921011,90332592148780928,7441355378176877032, %U A299141 612998553731816949526,50497150303834130394009,4159817600706531428357054 %N A299141 Number of nX7 0..1 arrays with every element equal to 1, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299141 Column 7 of A299142. %H A299141 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299141 Some solutions for n=5 %e A299141 ..0..0..0..1..1..0..1. .0..0..0..1..0..1..1. .0..0..0..0..0..1..1 %e A299141 ..0..0..1..1..0..1..0. .0..0..1..0..0..0..1. .0..0..1..0..1..0..1 %e A299141 ..0..0..0..1..1..0..1. .0..0..0..1..0..0..0. .0..0..0..1..1..0..1 %e A299141 ..0..0..0..0..0..0..1. .0..0..0..1..1..1..1. .0..0..0..1..1..0..1 %e A299141 ..0..0..1..1..0..0..0. .0..0..1..0..1..0..0. .0..0..1..0..1..0..0 %Y A299141 Cf. A299142. %K A299141 nonn,new %O A299141 1,1 %A A299141 _R. H. Hardin_, Feb 03 2018 %I A299140 %S A299140 5,888,43066,1850266,81988764,3643124959,161617794805,7170422794173, %T A299140 318133492126048,14114785587679567,626237253514871716, %U A299140 27784560111505472054,1232730529661461550436,54693130285484874044208 %N A299140 Number of nX6 0..1 arrays with every element equal to 1, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299140 Column 6 of A299142. %H A299140 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299140 Some solutions for n=5 %e A299140 ..0..1..1..0..1..1. .0..1..0..1..1..1. .0..0..0..1..1..0. .0..0..1..0..0..0 %e A299140 ..1..0..1..0..1..1. .1..0..1..1..0..0. .0..1..1..1..0..1. .1..1..1..0..1..0 %e A299140 ..0..0..0..0..0..1. .0..0..1..0..0..0. .0..0..1..1..0..0. .0..0..1..0..1..0 %e A299140 ..0..0..1..1..1..0. .0..0..0..1..0..1. .0..0..0..1..1..1. .0..0..0..1..1..1 %e A299140 ..0..0..0..0..1..0. .0..0..0..1..1..1. .0..0..0..0..0..1. .0..0..0..1..0..0 %Y A299140 Cf. A299142. %K A299140 nonn,new %O A299140 1,1 %A A299140 _R. H. Hardin_, Feb 03 2018 %I A299139 %S A299139 3,236,6205,143548,3426869,81988764,1958821107,46801360032, %T A299139 1118229413140,26717976955184,638375195180545,15252759731195845, %U A299139 364435654101670351,8707496129258580692,208049042144382510120,4970935766069111855568 %N A299139 Number of nX5 0..1 arrays with every element equal to 1, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299139 Column 5 of A299142. %H A299139 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299139 Some solutions for n=5 %e A299139 ..0..0..0..0..1. .0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0 %e A299139 ..0..0..0..1..0. .0..1..0..1..0. .0..0..0..1..1. .0..0..1..1..1 %e A299139 ..0..1..0..1..1. .1..0..1..0..1. .1..1..1..1..1. .1..1..1..0..1 %e A299139 ..1..1..1..1..1. .1..0..1..1..1. .0..0..0..0..1. .1..0..0..0..1 %e A299139 ..1..1..0..0..0. .1..0..0..0..0. .0..0..0..0..1. .0..0..0..1..1 %Y A299139 Cf. A299142. %K A299139 nonn,new %O A299139 1,1 %A A299139 _R. H. Hardin_, Feb 03 2018 %I A299138 %S A299138 2,64,899,11179,143548,1850266,23808476,306389599,3942948157, %T A299138 50742301057,653008378352,8403637443308,108147349359549, %U A299138 1391760325555663,17910719161925156,230495046446621416,2966266510949018995 %N A299138 Number of nX4 0..1 arrays with every element equal to 1, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299138 Column 4 of A299142. %H A299138 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299138 Empirical: a(n) = 12*a(n-1) +15*a(n-2) -33*a(n-3) -170*a(n-4) -418*a(n-5) -831*a(n-6) +290*a(n-7) +4188*a(n-8) +6777*a(n-9) +14211*a(n-10) +16195*a(n-11) +26645*a(n-12) -10951*a(n-13) -43014*a(n-14) +25645*a(n-15) +64072*a(n-16) -55702*a(n-17) -65256*a(n-18) -6799*a(n-19) +41979*a(n-20) +5773*a(n-21) -34173*a(n-22) -26481*a(n-23) -40556*a(n-24) -33028*a(n-25) -12415*a(n-26) +1169*a(n-27) +779*a(n-28) -497*a(n-29) -106*a(n-30) +80*a(n-31) +63*a(n-32) +14*a(n-33) for n>34 %e A299138 Some solutions for n=5 %e A299138 ..0..1..0..0. .0..0..1..0. .0..0..1..0. .0..0..0..1. .0..1..0..0 %e A299138 ..0..0..1..0. .0..1..0..0. .0..1..0..0. .0..1..1..1. .1..0..1..1 %e A299138 ..0..1..1..0. .1..1..0..1. .1..1..1..0. .0..0..1..0. .1..1..1..1 %e A299138 ..1..0..0..1. .0..0..1..0. .1..1..0..0. .0..0..1..0. .0..1..0..0 %e A299138 ..1..1..0..1. .0..0..1..0. .1..1..1..0. .0..0..1..0. .1..0..1..1 %Y A299138 Cf. A299142. %K A299138 nonn,new %O A299138 1,1 %A A299138 _R. H. Hardin_, Feb 03 2018 %I A299137 %S A299137 1,18,129,899,6205,43066,298361,2068149,14334327,99354814,688646455, %T A299137 4773147461,33083623049,229309139316,1589386941041,11016355018433, %U A299137 76356533603993,529242224053012,3668282443014641,25425590535640541 %N A299137 Number of nX3 0..1 arrays with every element equal to 1, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299137 Column 3 of A299142. %H A299137 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299137 Empirical: a(n) = 6*a(n-1) +8*a(n-2) -12*a(n-3) +8*a(n-4) +7*a(n-5) -4*a(n-6) -2*a(n-7) -7*a(n-8) -5*a(n-9) +2*a(n-10) for n>11 %e A299137 Some solutions for n=5 %e A299137 ..0..0..1. .0..0..0. .0..1..1. .0..1..1. .0..0..1. .0..1..1. .0..0..1 %e A299137 ..1..0..1. .0..1..0. .0..1..0. .0..0..1. .1..1..0. .1..0..0. .0..0..1 %e A299137 ..1..1..0. .0..0..1. .0..1..0. .1..0..0. .0..0..1. .0..1..1. .1..0..1 %e A299137 ..1..0..0. .1..1..0. .0..0..0. .1..0..1. .0..1..0. .0..0..1. .0..1..0 %e A299137 ..0..1..1. .1..0..1. .0..0..0. .1..1..0. .1..0..1. .1..1..1. .1..0..0 %Y A299137 Cf. A299142. %K A299137 nonn,new %O A299137 1,2 %A A299137 _R. H. Hardin_, Feb 03 2018 %I A299136 %S A299136 0,4,129,11179,3426869,3643124959,13311860331263,167720233672628013, %T A299136 7285451454600766025388 %N A299136 Number of nXn 0..1 arrays with every element equal to 1, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299136 Diagonal of A299142. %e A299136 Some solutions for n=5 %e A299136 ..0..0..0..0..1. .0..0..0..0..1. .0..0..0..0..0. .0..0..0..0..1 %e A299136 ..0..0..1..1..0. .0..1..0..1..0. .0..0..0..0..0. .0..0..0..1..1 %e A299136 ..0..1..1..0..0. .1..0..0..1..0. .0..1..0..0..1. .1..0..0..0..1 %e A299136 ..1..1..1..0..1. .1..0..1..1..0. .0..1..1..0..1. .1..1..1..0..1 %e A299136 ..0..0..1..1..0. .1..1..0..0..1. .1..1..1..0..1. .1..1..0..1..0 %Y A299136 Cf. A299142. %K A299136 nonn,new %O A299136 1,2 %A A299136 _R. H. Hardin_, Feb 03 2018 %I A299135 %S A299135 1,1,1,1,5,1,1,13,13,1,1,42,30,42,1,1,127,149,149,127,1,1,389,576, %T A299135 1261,576,389,1,1,1192,2621,9316,9316,2621,1192,1,1,3645,12495,75592, %U A299135 130924,75592,12495,3645,1,1,11161,59426,648807,1969223,1969223,648807,59426 %N A299135 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299135 Table starts %C A299135 .1.....1......1........1..........1.............1...............1 %C A299135 .1.....5.....13.......42........127...........389............1192 %C A299135 .1....13.....30......149........576..........2621...........12495 %C A299135 .1....42....149.....1261.......9316.........75592..........648807 %C A299135 .1...127....576.....9316.....130924.......1969223........31421743 %C A299135 .1...389...2621....75592....1969223......54173918......1576974771 %C A299135 .1..1192..12495...648807...31421743....1576974771.....83518793627 %C A299135 .1..3645..59426..5568411..497341456...45531416012...4394242516518 %C A299135 .1.11161.291819.48528385.7976508449.1330533860979.233738399148294 %H A299135 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299135 Empirical for column k: %F A299135 k=1: a(n) = a(n-1) %F A299135 k=2: a(n) = a(n-1) +5*a(n-2) +4*a(n-3) %F A299135 k=3: [order 12] for n>14 %F A299135 k=4: [order 49] for n>51 %e A299135 Some solutions for n=5 k=4 %e A299135 ..0..0..1..1. .0..1..1..0. .0..0..1..1. .0..0..1..1. .0..1..1..1 %e A299135 ..0..0..1..0. .1..1..1..1. .0..1..1..0. .0..0..1..1. .0..0..1..1 %e A299135 ..0..1..0..0. .0..0..0..1. .0..0..1..1. .1..1..0..0. .0..0..1..0 %e A299135 ..0..1..1..1. .0..0..0..0. .0..0..1..1. .1..1..0..0. .1..1..0..0 %e A299135 ..0..0..1..0. .1..0..0..0. .0..0..1..0. .0..1..0..0. .0..1..0..0 %Y A299135 Column 2 is A298234. %K A299135 nonn,tabl,new %O A299135 1,5 %A A299135 _R. H. Hardin_, Feb 03 2018 %I A299134 %S A299134 1,1192,12495,648807,31421743,1576974771,83518793627,4394242516518, %T A299134 233738399148294,12428256095957139,661754845010058955, %U A299134 35247766592323868205,1877651930930122311744,100034700318093782038164 %N A299134 Number of nX7 0..1 arrays with every element equal to 0, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299134 Column 7 of A299135. %H A299134 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299134 Some solutions for n=5 %e A299134 ..0..0..0..1..0..1..0. .0..0..1..1..1..1..1. .0..0..1..1..1..1..1 %e A299134 ..0..0..0..0..0..1..1. .0..0..1..0..0..1..0. .0..0..1..0..0..1..0 %e A299134 ..0..1..0..1..1..1..0. .0..0..1..0..1..0..0. .0..0..1..0..0..0..0 %e A299134 ..0..0..0..1..0..1..1. .0..0..1..0..0..1..1. .0..0..1..1..0..1..1 %e A299134 ..0..0..0..0..0..1..0. .0..0..1..1..1..1..1. .0..0..1..1..0..0..1 %Y A299134 Cf. A299135. %K A299134 nonn,new %O A299134 1,2 %A A299134 _R. H. Hardin_, Feb 03 2018 %I A299133 %S A299133 1,389,2621,75592,1969223,54173918,1576974771,45531416012, %T A299133 1330533860979,38865409949696,1136924948974306,33271000186874399, %U A299133 973756893397332773,28503033545959278420,834331965631542216193,24422884295729546061544 %N A299133 Number of nX6 0..1 arrays with every element equal to 0, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299133 Column 6 of A299135. %H A299133 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299133 Some solutions for n=5 %e A299133 ..0..0..1..1..0..1. .0..0..0..0..0..0. .0..0..0..1..1..0. .0..0..0..1..1..1 %e A299133 ..0..1..1..1..0..0. .1..0..1..1..0..1. .1..0..1..0..1..1. .0..0..0..1..1..0 %e A299133 ..0..0..1..0..0..1. .1..1..1..0..1..1. .1..1..1..1..0..0. .0..0..1..0..0..0 %e A299133 ..0..0..1..1..1..1. .0..0..1..1..0..1. .0..0..1..1..0..1. .0..0..0..1..1..0 %e A299133 ..0..0..1..0..1..1. .0..0..1..1..0..0. .0..0..0..0..0..0. .0..0..0..1..1..1 %Y A299133 Cf. A299135. %K A299133 nonn,new %O A299133 1,2 %A A299133 _R. H. Hardin_, Feb 03 2018 %I A299132 %S A299132 1,127,576,9316,130924,1969223,31421743,497341456,7976508449, %T A299132 127912383976,2054479924477,33013859983436,530587444370022, %U A299132 8528660855296842,137093294164198233,2203756183598761953 %N A299132 Number of nX5 0..1 arrays with every element equal to 0, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299132 Column 5 of A299135. %H A299132 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299132 Some solutions for n=5 %e A299132 ..0..0..1..0..0. .0..0..0..1..1. .0..0..0..1..1. .0..0..1..0..1 %e A299132 ..1..0..1..1..0. .0..0..0..1..1. .0..1..0..1..0. .0..0..1..1..1 %e A299132 ..0..0..1..1..1. .1..0..0..1..1. .1..1..1..0..0. .0..1..0..0..1 %e A299132 ..0..0..0..0..1. .1..1..0..1..1. .0..0..0..1..1. .1..0..0..0..1 %e A299132 ..1..0..1..0..0. .0..1..1..1..0. .0..0..0..0..1. .1..1..0..1..1 %Y A299132 Cf. A299135. %K A299132 nonn,new %O A299132 1,2 %A A299132 _R. H. Hardin_, Feb 03 2018 %I A299131 %S A299131 1,42,149,1261,9316,75592,648807,5568411,48528385,423758625, %T A299131 3708681524,32488998868,284725618163,2495882633803,21880813418029, %U A299131 191834630497049,1681904317156982,14746230774686806,129289560718813493 %N A299131 Number of nX4 0..1 arrays with every element equal to 0, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299131 Column 4 of A299135. %H A299131 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299131 Empirical: a(n) = 8*a(n-1) +32*a(n-2) -143*a(n-3) -887*a(n-4) +584*a(n-5) +9539*a(n-6) +11390*a(n-7) -45624*a(n-8) -94283*a(n-9) +79505*a(n-10) +237768*a(n-11) -357143*a(n-12) -646855*a(n-13) +2061269*a(n-14) +4087695*a(n-15) -2313416*a(n-16) -8006489*a(n-17) +1140920*a(n-18) +2127893*a(n-19) -20961100*a(n-20) -13259342*a(n-21) +35238300*a(n-22) +14771988*a(n-23) -45704036*a(n-24) -25008199*a(n-25) -18395747*a(n-26) +15237002*a(n-27) +4571949*a(n-28) +1400347*a(n-29) +95979339*a(n-30) -11078630*a(n-31) -7401783*a(n-32) -18283775*a(n-33) -68791478*a(n-34) +51991625*a(n-35) +45461324*a(n-36) -7931729*a(n-37) -17255775*a(n-38) -16853992*a(n-39) -6297911*a(n-40) +3467317*a(n-41) +2528188*a(n-42) -149388*a(n-43) +276392*a(n-44) -339084*a(n-45) -92248*a(n-46) +74064*a(n-47) -16048*a(n-48) +1248*a(n-49) for n>51 %e A299131 Some solutions for n=5 %e A299131 ..0..0..0..0. .0..1..1..1. .0..1..0..0. .0..0..0..1. .0..1..1..0 %e A299131 ..1..0..1..0. .1..1..0..1. .1..1..0..0. .0..1..0..0. .1..1..0..0 %e A299131 ..0..0..1..1. .0..0..0..0. .0..1..1..1. .1..1..1..0. .1..1..0..1 %e A299131 ..0..1..1..0. .0..0..1..0. .1..1..0..1. .0..1..1..1. .1..1..0..0 %e A299131 ..0..0..1..1. .0..1..1..1. .0..1..1..1. .1..1..1..1. .1..1..0..0 %Y A299131 Cf. A299135. %K A299131 nonn,new %O A299131 1,2 %A A299131 _R. H. Hardin_, Feb 03 2018 %I A299130 %S A299130 1,13,30,149,576,2621,12495,59426,291819,1434777,7089514,35139759, %T A299130 174262279,865022922,4295106017,21330145045,105941879250,526209580143, %U A299130 2613748830983,12983012681014,64489686157037,320337024303633 %N A299130 Number of nX3 0..1 arrays with every element equal to 0, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299130 Column 3 of A299135. %H A299130 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299130 Empirical: a(n) = 5*a(n-1) +7*a(n-2) -27*a(n-3) -65*a(n-4) +97*a(n-5) +118*a(n-6) -276*a(n-7) +110*a(n-8) +516*a(n-9) -226*a(n-10) -372*a(n-11) -12*a(n-12) for n>14 %e A299130 Some solutions for n=5 %e A299130 ..0..0..1. .0..0..1. .0..1..0. .0..0..1. .0..1..1. .0..1..1. .0..1..1 %e A299130 ..0..1..1. .1..0..0. .0..0..0. .0..1..1. .0..0..1. .0..0..1. .1..1..0 %e A299130 ..1..0..1. .0..0..1. .1..1..0. .1..1..1. .1..0..0. .0..1..1. .1..0..0 %e A299130 ..1..1..0. .1..0..0. .1..1..1. .1..0..0. .1..1..1. .0..0..0. .1..1..1 %e A299130 ..1..0..0. .1..1..0. .1..1..1. .0..0..1. .1..0..1. .1..0..0. .0..1..1 %Y A299130 Cf. A299135. %K A299130 nonn,new %O A299130 1,2 %A A299130 _R. H. Hardin_, Feb 03 2018 %I A299129 %S A299129 1,5,30,1261,130924,54173918,83518793627,421314290752028, %T A299129 7208059773839169432 %N A299129 Number of nXn 0..1 arrays with every element equal to 0, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299129 Diagonal of A299135. %e A299129 Some solutions for n=5 %e A299129 ..0..0..0..0..1. .0..0..0..1..1. .0..0..0..0..0. .0..0..1..0..0 %e A299129 ..0..0..1..0..0. .0..1..0..0..1. .0..1..1..0..0. .0..1..1..0..1 %e A299129 ..1..1..0..1..0. .1..1..1..1..0. .1..1..1..0..1. .0..0..0..1..1 %e A299129 ..0..1..1..1..1. .0..0..1..0..0. .1..1..1..0..0. .0..1..1..0..0 %e A299129 ..0..0..1..0..1. .0..0..0..0..0. .0..1..1..0..0. .0..0..1..0..1 %Y A299129 Cf. A299135. %K A299129 nonn,new %O A299129 1,2 %A A299129 _R. H. Hardin_, Feb 03 2018 %I A294671 %S A294671 3,6,4,10,2,7,10,14,13,9,7,16,12,7,17,14,6,13,14,12,0,10,7,15,11,13,8, %T A294671 12,9,3,10,8,16,14,2,10,13,9,10,9,12,8,9,11,12,14,9,8,7,14,6,13,7,9,7, %U A294671 3,14,6,14,16,16,9,7,12,13,10,0,12,5,2,13 %N A294671 Decimal expansion of the sum of sqrt(2) and sqrt(5) with no positional regrouping. %C A294671 a(n) is the sum of A002193(n) and A002163(n) without "carrying" (regrouping) when sum is greater than 9. %F A294671 a(n) = A002193(n) + A002163(n). %e A294671 for n = 8: A002193(8) = 5, A002163(8) = 9 -> a(8) = 14. %t A294671 Total[RealDigits[#, 10, 120][[1]] & /@ {Sqrt@ 2, Sqrt@ 5}] (* _Michael De Vlieger_, Nov 13 2017 *) %Y A294671 Cf. A002193, A002163, A073212, A294670. %K A294671 base,easy,nonn,new %O A294671 1,1 %A A294671 _Lyle Blosser_, Nov 06 2017 %I A294670 %S A294670 3,6,5,0,2,8,1,5,3,9,8,7,2,8,8,4,7,4,5,2,1,0,8,6,2,3,9,2,9,4,0,9,7,4, %T A294670 3,1,4,0,1,0,2,9,0,2,3,4,9,8,8,4,7,3,7,9,7,4,4,7,5,7,6,9,8,3,4,0,1,2, %U A294670 5,3,4,0,4,0,9,9,9,1,1,9,3,8,2,6 %N A294670 Decimal expansion of the sum sqrt(2) + sqrt(5). %F A294670 Equals A002193 + A002163. %e A294670 3.65028153987288474521086239294... %p A294670 evalf(sqrt(2)+sqrt(5),200); # _Wesley Ivan Hurt_, Nov 07 2017 %t A294670 RealDigits[Sqrt@ 2 + Sqrt@ 5, 10, 120][[1]] (* _Michael De Vlieger_, Nov 13 2017 *) %Y A294670 Cf. A002193 (sqrt(2)), A002163 (sqrt(5)), A172323, A294671. %K A294670 cons,easy,nonn,new %O A294670 1,1 %A A294670 _Lyle Blosser_, Nov 06 2017 %I A295041 %S A295041 0,1,0,1,3,0,2,1,5,3,4,0,7,2,6,1,9,5,8,3,11,4,10,0,13,7,12,2,15,6,14, %T A295041 1,17,9,16,5,19,8,18,3,21,11,20,4,23,10,22,0,25,13,24 %N A295041 The Grundy number of restricted Nim with a pass move. %C A295041 These are the Grundy values or nim-values for heaps of n beans in the game where you're allowed to take up to half of the beans in a heap and you can use a one-time pass, i.e., a pass move which may be used at most once in a game, and not from a terminal position. Once the pass has been used by either player, it is no longer available. If the pass move were not allowed, then this game would be the same as the one in A025480. %F A295041 a(4k) = 2k+1; a(4k+2) = 2k; a(4k+3) = a(2k+1); a(8k+1) = 2k+1; a(8k+5) = 2k. %t A295041 f[n_] := Which[IntegerQ[n/4], (n + 2)/2, IntegerQ[(n - 2)/4], (n - 2)/2, %t A295041 IntegerQ[(n - 3)/4], f[(n - 1)/2], IntegerQ[(n - 1)/8], (n + 3)/4, %t A295041 IntegerQ[(n - 5)/8], (n - 5)/4]; %t A295041 (* the following is Mathematica program to generate the same sequence as Grundy numbers *) %t A295041 ss = 50; allcases = Flatten[Table[Table[{a, pass}, {a, 0, ss}], {pass, 0, 1}], 1]; %t A295041 move[z_] := Block[{p}, p = z; %t A295041 a = p[[1]]; pass = p[[2]]; c0 = Floor[a/2]; %t A295041 Which[a > 0 && pass == 1, %t A295041 Union[Table[{a - x, pass}, {x, 1, c0}], {{a, 0}}], a > 0, %t A295041 Table[{a - x, pass}, {x, 1, c0}], a == 0, {}]]; %t A295041 Mex[L_] := Min[Complement[Range[0, Length[L]], L]]; %t A295041 Gr2[pos_] := Gr2[pos] = Mex[Map[Gr2, move[pos]]]; %t A295041 pposition = Select[allcases, Gr2[#] == 0 &]; %t A295041 Table[Gr2[{n, 1}], {n, 0, 50}] %Y A295041 Cf. A025480. %K A295041 nonn,new %O A295041 0,5 %A A295041 _Ryouhei Miyadera_, Mariko Kashihara and Koh Oomori, Nov 12 2012 %I A299128 %S A299128 1,2,2,3,4,3,5,4,4,5,8,16,9,16,8,13,50,23,23,50,13,21,112,31,232,31, %T A299128 112,21,34,348,184,623,623,184,348,34,55,1028,427,3368,2636,3368,427, %U A299128 1028,55,89,2796,1115,20226,13809,13809,20226,1115,2796,89,144,8216,4128,95305 %N A299128 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299128 Table starts %C A299128 ..1....2....3......5.......8........13..........21...........34.............55 %C A299128 ..2....4....4.....16......50.......112.........348.........1028...........2796 %C A299128 ..3....4....9.....23......31.......184.........427.........1115...........4128 %C A299128 ..5...16...23....232.....623......3368.......20226........95305.........517446 %C A299128 ..8...50...31....623....2636.....13809......132409.......854671........5933446 %C A299128 .13..112..184...3368...13809....198753.....2286535.....21688754......284403950 %C A299128 .21..348..427..20226..132409...2286535....50126162....761448901....14747885320 %C A299128 .34.1028.1115..95305..854671..21688754...761448901..17703574257...537784640291 %C A299128 .55.2796.4128.517446.5933446.284403950.14747885320.537784640291.28274774976231 %H A299128 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299128 Empirical for column k: %F A299128 k=1: a(n) = a(n-1) +a(n-2) %F A299128 k=2: a(n) = 2*a(n-1) +2*a(n-2) +6*a(n-3) -10*a(n-4) -8*a(n-5) for n>6 %F A299128 k=3: [order 18] for n>19 %F A299128 k=4: [order 66] for n>68 %e A299128 Some solutions for n=5 k=4 %e A299128 ..0..1..1..0. .0..1..1..0. .0..0..1..1. .0..1..1..0. .0..0..1..1 %e A299128 ..0..1..1..0. .0..1..1..0. .0..0..1..1. .0..1..1..1. .1..1..1..1 %e A299128 ..0..0..1..1. .1..1..0..0. .0..0..1..1. .0..0..1..0. .1..1..0..0 %e A299128 ..0..0..1..0. .0..1..0..0. .1..0..1..0. .1..0..1..1. .1..1..1..1 %e A299128 ..1..0..0..1. .0..1..0..0. .1..0..1..0. .1..0..1..1. .0..1..0..0 %Y A299128 Column 1 is A000045(n+1). %Y A299128 Column 2 is A298148. %K A299128 nonn,tabl,new %O A299128 1,2 %A A299128 _R. H. Hardin_, Feb 03 2018 %I A297103 %S A297103 1,1,1,2,5,7,10,20,41,67,110,220,441,767,1335,2670,5341,9587,17211, %T A297103 34422,68845,126011,230655,461310,922621,1711595,3175311,6350622, %U A297103 12701245,23796515,44584536,89169072,178338145 %N A297103 The number of equal-sized squares in the highest stack of squares contained in successive Genealodrons formed from 2^n - 1 equal-sized squares. %C A297103 The first Genealodron consists of one square. %C A297103 The second Genealodron is formed by joining another equal-sized square to the left edge and to the right edge of the first so that the second Genealodron is made up of three squares. %C A297103 The third Genealodron is formed by joining squares to the upper and lower edges of both the second and third square of the second Genealodron so that the third Genealodron is made up of seven squares. %C A297103 The fourth Genealodron is formed by joining squares to the left and right edges of the fourth, fifth, sixth and seventh squares of the third Genealodron so that the fourth Genealodron has fifteen squares. The fourth Genealodron has the first overlaps, so although it contains 15 squares only 13 are seen when it is viewed from above. %C A297103 The fifth Genealodron is formed by adding 16 more squares to the upper and lower edges of the last eight squares added to the fourth Genealodron so the fifth Genealodron has 31 squares, only 21 of which are seen when it is viewed from above because of the increasing number of overlaps. %C A297103 The sixth Genealodron is formed by adding 32 more squares to the left and right edge of the last 16 squares added to the fifth Genealodron. So the sixth Genealodron has 63 squares only 31 of which are visible. %C A297103 This continues, and the edges on which the new squares are added keep alternating between left and right and then upper and lower. %C A297103 Gradually within the Genealodron, spirals are building counterclockwise and clockwise. The sequence that the Genealodron built with squares generates is different from the one built with equilateral triangles, because when a square is added, the spiral then turns through 90 degrees rather than just 60 degrees. %H A297103 Andrew Smith, Illustration of initial terms %o A297103 (MATLAB) %o A297103 %I solved the problem by representing each Genealodron as a matrix %o A297103 n=input('how many terms?'); %o A297103 %preallocation of length of output (length n) %o A297103 vec=zeros(1,n); %o A297103 %below I initialize the first 3 terms which are easily done with pen and paper %o A297103 vec(1)=1; %o A297103 vec(2)=1; %o A297103 vec(3)=1; %o A297103 %imat is the intermediate matrix to go from 3rd to 4th matrix. %o A297103 imat=[1,0,1;0,0,0;1,0,1]; %o A297103 %mat is the 3rd matrix %o A297103 mat=[1,0,1;1,1,1;1,0,1]; %o A297103 %loop %o A297103 for i=4:n %o A297103 %when i is even %o A297103 if mod(i,2)==0 %o A297103 imat2=[zeros(i-1,2),imat]; %o A297103 imat3=[imat,zeros(i-1,2)]; %o A297103 %superposing two variations of previous intermediate matrix to get next one %o A297103 imat=imat2+imat3; %o A297103 %making mat same size as imat %o A297103 mat=[zeros(i-1,1),mat,zeros(i-1,1)]; %o A297103 %calculating new matrix (=old matrix+intermediate) %o A297103 mat=mat+imat; %o A297103 %similarly when i is odd %o A297103 else %o A297103 imat2=[zeros(2,i);imat]; %o A297103 imat3=[imat;zeros(2,i)]; %o A297103 imat=imat2+imat3; %o A297103 mat=[zeros(1,i);mat;zeros(1,i)]; %o A297103 mat=mat+imat; %o A297103 end %o A297103 %working out the maximum value of new matrix and allocating it to a position in the output vector %o A297103 vec(i)=max(max(mat)); %o A297103 end %o A297103 format long g %o A297103 disp(vec) %Y A297103 Cf. A179178, A179316. %K A297103 nonn,easy,new %O A297103 1,4 %A A297103 _Andrew Smith_, Dec 25 2017 %I A299127 %S A299127 21,348,427,20226,132409,2286535,50126162,761448901,14747885320, %T A299127 292184974732,5319449303150,102312136544858,1991770170247881, %U A299127 37605783844654553,722670947723793458,13939476383806855997 %N A299127 Number of nX7 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299127 Column 7 of A299128. %H A299127 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299127 Some solutions for n=5 %e A299127 ..0..1..1..0..0..1..0. .0..0..0..1..1..1..0. .0..0..0..1..0..1..1 %e A299127 ..1..0..1..1..1..1..1. .0..0..0..1..1..1..0. .0..0..0..0..0..0..0 %e A299127 ..0..0..1..0..1..1..0. .0..0..1..0..0..0..0. .1..1..1..1..1..0..0 %e A299127 ..1..0..0..1..1..1..1. .0..0..0..0..1..0..0. .1..1..0..0..1..1..0 %e A299127 ..0..1..0..0..0..1..0. .1..0..1..1..1..1..1. .0..0..0..0..1..1..0 %Y A299127 Cf. A299128. %K A299127 nonn,new %O A299127 1,1 %A A299127 _R. H. Hardin_, Feb 03 2018 %I A299126 %S A299126 13,112,184,3368,13809,198753,2286535,21688754,284403950,3307001257, %T A299126 37294174386,457492833765,5410790816292,63288087145558, %U A299126 760502020473486,9023717299845918,106698982952917563,1272922168774272137 %N A299126 Number of nX6 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299126 Column 6 of A299128. %H A299126 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299126 Some solutions for n=5 %e A299126 ..0..0..1..1..0..0. .0..1..0..1..0..0. .0..0..0..1..0..0. .0..1..1..1..0..1 %e A299126 ..0..0..0..0..0..0. .1..0..0..0..0..0. .1..1..0..0..0..0. .1..0..1..1..0..1 %e A299126 ..1..1..0..1..1..1. .1..1..1..0..1..1. .0..0..0..1..1..0. .1..1..1..1..0..0 %e A299126 ..0..0..0..1..1..1. .1..0..1..1..1..1. .1..1..0..0..0..0. .0..1..0..0..0..0 %e A299126 ..0..0..1..0..1..1. .1..0..1..1..1..1. .1..1..1..1..0..0. .0..1..0..0..1..1 %Y A299126 Cf. A299128. %K A299126 nonn,new %O A299126 1,1 %A A299126 _R. H. Hardin_, Feb 03 2018 %I A299125 %S A299125 8,50,31,623,2636,13809,132409,854671,5933446,48107376,347763463, %T A299125 2553887073,19707474302,146173594966,1090822062274,8266516495684, %U A299125 61810826216746,463349163999396,3489651468586305,26165253287394361 %N A299125 Number of nX5 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299125 Column 5 of A299128. %H A299125 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299125 Some solutions for n=5 %e A299125 ..0..1..0..1..0. .0..1..0..0..1. .0..0..0..0..0. .0..1..1..1..1 %e A299125 ..1..0..0..0..1. .0..1..0..0..1. .0..0..1..0..0. .1..0..1..0..0 %e A299125 ..1..1..0..1..1. .1..1..1..1..1. .0..0..0..0..0. .1..1..1..0..0 %e A299125 ..1..1..1..1..1. .0..1..0..0..1. .0..1..1..1..0. .0..1..0..0..1 %e A299125 ..1..1..1..1..1. .1..0..0..1..0. .1..0..1..1..0. .1..0..0..1..0 %Y A299125 Cf. A299128. %K A299125 nonn,new %O A299125 1,1 %A A299125 _R. H. Hardin_, Feb 03 2018 %I A299124 %S A299124 5,16,23,232,623,3368,20226,95305,517446,2849177,14996918,80878846, %T A299124 438307578,2350137185,12666447836,68352017686,367891360100, %U A299124 1982533944388,10686608415147,57568972855703,310217349581539,1671740619224755 %N A299124 Number of nX4 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299124 Column 4 of A299128. %H A299124 R. H. Hardin, Table of n, a(n) for n = 1..210 %H A299124 R. H. Hardin, Empirical recurrence of order 66 %F A299124 Empirical recurrence of order 66 (see link above) %e A299124 Some solutions for n=5 %e A299124 ..0..0..0..0. .0..1..0..0. .0..0..1..0. .0..0..1..0. .0..1..0..0 %e A299124 ..0..0..1..1. .1..0..0..0. .0..0..1..0. .0..0..0..1. .0..1..0..0 %e A299124 ..0..0..0..0. .1..1..1..1. .0..1..1..1. .1..1..0..0. .1..1..0..0 %e A299124 ..1..1..1..1. .1..1..0..0. .0..0..1..1. .1..1..0..0. .0..1..0..0 %e A299124 ..1..1..1..1. .1..1..1..1. .0..0..0..0. .0..0..0..1. .0..1..0..0 %Y A299124 Cf. A299128. %K A299124 nonn,new %O A299124 1,1 %A A299124 _R. H. Hardin_, Feb 03 2018 %I A299123 %S A299123 3,4,9,23,31,184,427,1115,4128,11024,32827,104959,302093,917104, %T A299123 2804565,8324555,25231517,76267810,228857830,691699768,2085676297, %U A299123 6280152762,18952185100,57128981249,172189537544,519305913429,1565482526696 %N A299123 Number of nX3 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299123 Column 3 of A299128. %H A299123 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299123 Empirical: a(n) = a(n-1) +4*a(n-2) +16*a(n-3) -13*a(n-4) -35*a(n-5) -62*a(n-6) +28*a(n-7) +65*a(n-8) +36*a(n-9) +61*a(n-10) +59*a(n-11) -34*a(n-12) -64*a(n-13) -36*a(n-14) -40*a(n-15) +3*a(n-16) +18*a(n-17) +2*a(n-18) for n>19 %e A299123 Some solutions for n=5 %e A299123 ..0..0..0. .0..1..0. .0..1..1. .0..0..0. .0..0..0. .0..1..1. .0..1..0 %e A299123 ..0..0..0. .1..1..1. .0..1..1. .0..0..0. .0..0..0. .1..1..1. .0..1..0 %e A299123 ..0..1..0. .1..0..1. .0..0..0. .0..0..1. .0..1..0. .1..1..1. .0..0..0 %e A299123 ..0..0..0. .1..1..1. .0..1..1. .0..0..0. .0..0..0. .0..0..0. .1..0..1 %e A299123 ..0..0..0. .0..1..0. .0..1..1. .0..0..0. .1..0..1. .0..0..0. .1..0..1 %Y A299123 Cf. A299128. %K A299123 nonn,new %O A299123 1,1 %A A299123 _R. H. Hardin_, Feb 03 2018 %I A299122 %S A299122 1,4,9,232,2636,198753,50126162,17703574257,28274774976231 %N A299122 Number of nXn 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299122 Diagonal of A299128. %e A299122 Some solutions for n=5 %e A299122 ..0..1..1..1..1. .0..1..1..1..0. .0..1..1..1..1. .0..1..1..0..0 %e A299122 ..1..0..1..1..1. .0..1..1..1..1. .0..1..1..0..0. .0..1..1..1..1 %e A299122 ..0..0..1..0..1. .0..0..1..1..1. .0..0..0..0..0. .0..0..1..0..0 %e A299122 ..1..0..0..1..1. .1..0..1..0..1. .1..0..1..1..1. .0..0..1..0..0 %e A299122 ..1..0..0..1..1. .0..1..1..1..0. .0..1..1..1..1. .1..1..1..1..1 %Y A299122 Cf. A299128. %K A299122 nonn,new %O A299122 1,2 %A A299122 _R. H. Hardin_, Feb 03 2018 %I A299121 %S A299121 1,2,3,6,38,1490,443762,965262242,23539096637282,4878608938121399042, %T A299121 16752209028723653862101762,531115497554502361264846265433602, %U A299121 392660148984369152453298787770243889113602,2811644066816242246665284589590844386691155533363202 %N A299121 a(n) = Sum_{k=0..n} (k*(n-k))!. %F A299121 a(n) ~ sqrt(Pi) * n^(n^2/2 + 1) / (2^(n^2/2 + 1/2) * exp(n^2/4)) if n is even, and a(n) ~ sqrt(Pi) * n^(n^2/2 + 1/2) / (2^(n^2/2 - 1) * exp(n^2/4)) if n is odd. %t A299121 Table[Sum[(k*(n-k))!, {k, 0, n}], {n, 0, 14}] %Y A299121 Cf. A000142, A003149, A003422. %K A299121 nonn,new %O A299121 0,2 %A A299121 _Vaclav Kotesovec_, Feb 03 2018 %I A299108 %S A299108 1,1,3,9,27,79,231,675,1971,5755,16805,49071,143289,418411,1221781, %T A299108 3567663,10417761,30420401,88829145,259385701,757419669,2211704625, %U A299108 6458291945,18858546645,55067931981,160801210705,469547855419,1371104033121,4003694720243 %N A299108 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)). %H A299108 Vaclav Kotesovec, Table of n, a(n) for n = 0..2000 %H A299108 N. J. A. Sloane, Transforms %F A299108 G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)). %F A299108 G.f.: 1/(1 - x/theta_4(x)), where theta_4() is the Jacobi theta function. %F A299108 a(0) = 1; a(n) = Sum_{k=1..n} A015128(k-1)*a(n-k). %F A299108 a(n) ~ c * d^n, where d = 2.9200517419026569743994130834319365190407162724411912701937027582419975778... is the root of the equation EllipticTheta(4, 0, 1/d) * d = 1 and c = 0.372842695601022868809531452599286285949969156503576039087883242107... - _Vaclav Kotesovec_, Feb 03 2018 %p A299108 S:= series(1/(1-x/JacobiTheta4(0,x)),x,51): %p A299108 seq(coeff(S,x,n),n=0..50); # _Robert Israel_, Feb 02 2018 %t A299108 nmax = 28; CoefficientList[Series[1/(1 - x Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x] %t A299108 nmax = 28; CoefficientList[Series[1/(1 - x/EllipticTheta[4, 0, x]), {x, 0, nmax}], x] %t A299108 nmax = 28; CoefficientList[Series[1/(1 - x QPochhammer[-x, x]/QPochhammer[x, x]), {x, 0, nmax}], x] %Y A299108 Antidiagonal sums of A288515. %Y A299108 Cf. A015128, A032803, A067687, A299105, A299106. %K A299108 nonn,new %O A299108 0,3 %A A299108 _Ilya Gutkovskiy_, Feb 02 2018 %I A299106 %S A299106 1,1,2,4,9,19,41,88,189,405,869,1864,3998,8575,18392,39448,84610, %T A299106 181475,389235,834848,1790617,3840591,8237462,17668057,37895195, %U A299106 81279216,174331098,373912708,801983781,1720128713,3689404772,7913191304,16972547194,36403436640 %N A299106 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)). %H A299106 Vaclav Kotesovec, Table of n, a(n) for n = 0..3000 %H A299106 N. J. A. Sloane, Transforms %F A299106 G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)). %F A299106 a(0) = 1; a(n) = Sum_{k=1..n} A000009(k-1)*a(n-k). %F A299106 a(n) ~ c * d^n, where d = 2.14484226934608840026733598736202689102117985119507858808036465196716739... is the root of the equation QPochhammer(1/d, 1/d^2)*d = 1 and c = 0.4217892515709863296976217395517853732959704351198250451894928058439... - _Vaclav Kotesovec_, Feb 03 2018 %t A299106 nmax = 33; CoefficientList[Series[1/(1 - x Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x] %t A299106 nmax = 33; CoefficientList[Series[1/(1 - x/QPochhammer[x, x^2]), {x, 0, nmax}], x] %t A299106 a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}] %Y A299106 Antidiagonal sums of A286335. %Y A299106 Cf. A000009, A067687, A299105, A299108. %K A299106 nonn,new %O A299106 0,3 %A A299106 _Ilya Gutkovskiy_, Feb 02 2018 %I A299112 %S A299112 1,2,6,12,16,30,42,54,66,78,105,120,156,330,525,546,831,1071,1251, %T A299112 1386,1875,1890,2136,2241,2331,2541,2565,2736,2871,3165,3606,3885,4710 %N A299112 Record values in A243145. %p A299112 f:= proc(n) local k; %p A299112 for k from (n mod 2)+1 by 2 do %p A299112 if isprime(n+k) and isprime(n+k^2) then return k fi %p A299112 od %p A299112 end proc: %p A299112 f(1):= 1: %p A299112 vmax:= 0: recvals:= NULL: %p A299112 for n from 1 to 10^7 do %p A299112 v:= f(n); %p A299112 if v > vmax then %p A299112 vmax:= v; recvals:= recvals, v; %p A299112 fi %p A299112 od: %p A299112 recvals; %Y A299112 Cf. A243145, A299073. %K A299112 nonn,more,new %O A299112 1,2 %A A299112 _Robert Israel_, Feb 02 2018 %I A299073 %S A299073 1,3,5,19,21,41,59,173,241,269,326,341,431,491,4406,6641,10946,45386, %T A299073 54206,81611,94706,362351,486281,1099766,1197026,1220126,1439756, %U A299073 1597361,1685276,2000246,2405051,6550946,9996731 %N A299073 Indices of records in A243145. %p A299073 f:= proc(n) local k; %p A299073 for k from (n mod 2)+1 by 2 do %p A299073 if isprime(n+k) and isprime(n+k^2) then return k fi %p A299073 od %p A299073 end proc: %p A299073 f(1):= 1: %p A299073 vmax:= 0: recinds:= NULL: %p A299073 for n from 1 to 10^7 do %p A299073 v:= f(n); %p A299073 if v > vmax then %p A299073 vmax:= v; recinds:= recinds, n; %p A299073 fi %p A299073 od: %p A299073 recinds; %Y A299073 Cf. A243145, A299112. %K A299073 nonn,more,new %O A299073 1,2 %A A299073 _Robert Israel_, Feb 02 2018 %I A299113 %S A299113 1,1,3,12,52,247,1226,6299,33209,178618,976296,5407384,30283120, %T A299113 171196956,975662480,5599508648,32334837886,187737500013, %U A299113 1095295264857,6417886638389,37752602033079,222861754454841,1319834477009635,7839314017612273,46688045740233741 %N A299113 Number of rooted identity trees with 2n+1 nodes. %H A299113 Alois P. Heinz, Table of n, a(n) for n = 0..1253 %F A299113 a(n) = A004111(2n+1). %e A299113 a(2) = 3: %e A299113 o o o %e A299113 | | / \ %e A299113 o o o o %e A299113 | / \ | %e A299113 o o o o %e A299113 | | | %e A299113 o o o %e A299113 | %e A299113 o %p A299113 with(numtheory): %p A299113 b:= proc(n) option remember; `if`(n<2, n, add(b(n-k)*add( %p A299113 b(d)*d*(-1)^(k/d+1), d=divisors(k)), k=1..n-1)/(n-1)) %p A299113 end: %p A299113 a:= n-> b(2*n+1): %p A299113 seq(a(n), n=0..30); %Y A299113 Bisection of A004111 (odd part). %Y A299113 Cf. A100427, A299098. %K A299113 nonn,new %O A299113 0,3 %A A299113 _Alois P. Heinz_, Feb 02 2018 %I A299098 %S A299098 0,1,2,6,25,113,548,2770,14426,76851,416848,2294224,12780394,71924647, %T A299098 408310668,2335443077,13446130438,77863375126,453203435319, %U A299098 2649957419351,15558520126830,91687179000949,542139459641933,3215484006733932,19125017153077911 %N A299098 Number of rooted identity trees with 2n nodes. %H A299098 Alois P. Heinz, Table of n, a(n) for n = 0..1253 %F A299098 a(n) = A004111(2n). %e A299098 a(3) = 6: %e A299098 o o o o o o %e A299098 | | | / \ / \ / \ %e A299098 o o o o o o o o o %e A299098 | | / \ | | | / \ %e A299098 o o o o o o o o o %e A299098 | / \ | | | | %e A299098 o o o o o o o %e A299098 | | | | %e A299098 o o o o %e A299098 | %e A299098 o %p A299098 with(numtheory): %p A299098 b:= proc(n) option remember; `if`(n<2, n, add(b(n-k)*add( %p A299098 b(d)*d*(-1)^(k/d+1), d=divisors(k)), k=1..n-1)/(n-1)) %p A299098 end: %p A299098 a:= n-> b(2*n): %p A299098 seq(a(n), n=0..30); %Y A299098 Bisection of A004111 (even part). %Y A299098 Cf. A100034, A299039, A299113. %K A299098 nonn,new %O A299098 0,3 %A A299098 _Alois P. Heinz_, Feb 02 2018 %I A298801 %S A298801 4,3,6,0,1,2,5,9,12,7,8,15,10,11,18,13,14,21,16,17,24,19,20,27,22,23, %T A298801 30,25,26,33,28,29,36,31,32,39,34,35,42,37,38,45,40,41,48,43,44,51,46, %U A298801 47,54,49,50,57,52,53,60,55,56,63,58,59,66,61,62,69,64,65,72,67,68,75,70,71,78,73,74,81,76 %N A298801 Fourth column of triangular array in A296339. %C A298801 This was the first column of A296339 for which no simple formula was known (cf. A004483, A004482). (Since these are Grundy values for a certain game, there is a complicated recurrence involving the whole triangle.) The formula below matches the data, and is fairly short (but ugly). %F A298801 It appears that for n >= 8, a(n) = tersum(n,1) + 6 if n == 2 (mod 3), otherwise tersum(n,1) - 3. %F A298801 Conjectures from _Colin Barker_, Feb 03 2018: (Start) %F A298801 G.f.: (4 - x + 3*x^2 - 10*x^3 + 2*x^4 - 2*x^5 + 9*x^6 + 3*x^7 + 2*x^8 - 8*x^9 - 3*x^10 + 4*x^11) / ((1 - x)^2*(1 + x + x^2)). %F A298801 a(n) = a(n-1) + a(n-3) - a(n-4) for n>11. %F A298801 (End) %Y A298801 Cf. A296339, A004482, A004483, A296340. %K A298801 nonn,new %O A298801 0,1 %A A298801 _N. J. A. Sloane_, Feb 02 2018 %I A299103 %S A299103 73,433,601,673,1801,4513,18433,32377,37633,54001,55201,61681,63901, %T A299103 66529,100801,115201,121369,122921,168781,178481,187417,203617,210913, %U A299103 258721,286721,370661,414721,588061,649657,695701,737537,1781921,3194101,4674797,4681801,5039581,6433561,7593961,7692697 %N A299103 Primes p = x^2 + y^2, not of form x^2 + 1, such that 2^(x^2) == 1 (mod p) or 2^(y^2) == 1 (mod p). %C A299103 Primes p = x^2 + y^2, not of form x^2 + 1, such that 2^(x^2) == 2 (mod p) or 2^(y^2) == 2 (mod p). %H A299103 Max Alekseyev, Table of n, a(n) for n = 1..100 %e A299103 For 73 = 3^2 + 8^2 we have 2^9 == 1 (mod 73) and 2^64 == 2 (mod 73). %p A299103 f:= proc(p) local F,x,y; %p A299103 if not isprime(p) then return false fi; %p A299103 if issqr(p-1) then return false fi; %p A299103 F:= GaussInt:-GIfactors(p)[2]; %p A299103 x,y:= (Re,Im)(F[1][1]); %p A299103 2 &^ (x^2) mod p = 1 or 2 &^ (y^2) mod p = 1 %p A299103 end proc: %p A299103 select(f, [seq(i,i=5..10^7,4)]); # _Robert Israel_, Feb 02 2018 %o A299103 (PARI) B=bnfinit(x^2+1); { is_A299103(p) = my(z); if(p%4!=1 || issquare(p-1),return(0)); z=abs(Vec(bnfisintnorm(B,p)[1])); Mod(2,p)^(z[1]^2)==1 || Mod(2,p)^(z[2]^2)==1; } \\ _Max Alekseyev_, Feb 02 2018 %Y A299103 Cf. A002313, A002496. %K A299103 nonn,new %O A299103 1,1 %A A299103 _Thomas Ordowski_, Feb 02 2018 %E A299103 Terms a(7) onward from _Max Alekseyev_, Feb 02 2018 %I A299105 %S A299105 1,1,0,-2,-3,-1,5,10,7,-9,-29,-30,10,77,108,22,-184,-351,-207,372, %T A299105 1041,969,-516,-2835,-3655,-284,6990,12190,5977,-14957,-37044,-30994, %U A299105 24144,103374,122409,-7715,-262704,-420585,-162274,589068,1309674,972747,-1057935,-3742955 %N A299105 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)). %H A299105 N. J. A. Sloane, Transforms %F A299105 G.f.: 1/(1 - x*Product_{k>=1} (1 - x^k)). %F A299105 a(0) = 1; a(n) = Sum_{k=1..n} A010815(k-1)*a(n-k). %t A299105 nmax = 43; CoefficientList[Series[1/(1 - x Product[1 - x^k, {k, 1, nmax}]), {x, 0, nmax}], x] %t A299105 nmax = 43; CoefficientList[Series[1/(1 - x QPochhammer[x, x]), {x, 0, nmax}], x] %Y A299105 Antidiagonal sums of A286354. %Y A299105 Cf. A010815, A067687, A299106, A299108. %K A299105 sign,new %O A299105 0,4 %A A299105 _Ilya Gutkovskiy_, Feb 02 2018 %I A295811 %S A295811 1,1,2,11,140,2898,80844,2786091,113184008,5266198778,275248731860, %T A295811 15939117549502,1012084698990904,69901132180300132, %U A295811 5217426460077854712,418615099531669351443,35942031310982080239120,3289533291926922095871546,319841125714352173292953668,32937612567848507536114539402,3582858531960091228861488651864 %N A295811 G.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2) = [x^(n-2)] 2*n*A(x)^(n^2) for n>=2, with A(0) = 1. %C A295811 Compare g.f. to: [x^(n-1)] G(x)^n = [x^(n-2)] 2*G(x)^n for n>=2 holds when G(x) = 1/(1-x). %H A295811 Paul D. Hanna, Table of n, a(n) for n = 0..300 %F A295811 a(2^k - 1) is odd for k>=0 and a(n) is even elsewhere (conjecture). %F A295811 a(n) ~ c * d^n * n! / n^3, where d = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.176554609483480358231680164050876553672889794284... and c = 2.719099850893334482... - _Vaclav Kotesovec_, Feb 07 2018 %e A295811 G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 140*x^4 + 2898*x^5 + 80844*x^6 + 2786091*x^7 + 113184008*x^8 + 5266198778*x^9 + 275248731860*x^10 + ... %e A295811 ILLUSTRATION OF THE DEFINITION. %e A295811 The table of coefficients of x^k in A(x)^(n^2) begins: %e A295811 n=1: [1, 1, 2, 11, 140, 2898, 80844, ...]; %e A295811 n=2: [1, 4, 14, 72, 741, 13724, 364546, ...]; %e A295811 n=3: [1, 9, 54, 327, 2826, 42660, 1017720, ...]; %e A295811 n=4: [1, 16, 152, 1216, 10540, 129376, 2559792, ...]; %e A295811 n=5: [1, 25, 350, 3775, 37750, 427480, 6820800, ...]; %e A295811 n=6: [1, 36, 702, 10056, 123165, 1477980, 20712546, ...]; %e A295811 n=7: [1, 49, 1274, 23667, 359856, 4953998, 69355972, ...]; ... %e A295811 in which the main diagonal %e A295811 D0 = [1, 4, 54, 1216, 37750, 1477980, 69355972, 3775816704, ...] %e A295811 and the adjacent diagonal %e A295811 D1 = [1, 9, 152, 3775, 123165, 4953998, 235988544, 12954335103, ...] %e A295811 are related by D0[n-1] = 2*n*D1[n-2] for n>=2. %e A295811 The related sequence D0[n-1]/n^2, n>=1, begins: %e A295811 [1, 1, 6, 76, 1510, 41055, 1415428, 58997136, 2878741134, 160698224230, ...]. %o A295811 (PARI) {a(n) = my(A=[1]); for(m=2, n+1, A=concat(A, 0); V=Vec(Ser(A)^(m^2)); A[#A] = V[#A-1]*2/m - V[#A]/m^2 ); A[n+1]} %o A295811 for(n=0,20,print1(a(n),", ")) %Y A295811 Cf. A295766, A088715. %K A295811 nonn,new %O A295811 0,3 %A A295811 _Paul D. Hanna_, Feb 02 2018 %I A298800 %S A298800 2,2,14,2,70,184,2,118,648,256,2,198,1656,2240 %N A298800 Triangle read by rows: T(n,k) = number of Ringel ladders of order n and genus k. %D A298800 Tesar, Esther Hunt. "Genus distribution of Ringel ladders." Discrete Mathematics 216.1-3 (2000): 235-252. %e A298800 Triangle begins: %e A298800 2, %e A298800 2,14, %e A298800 2,70,184, %e A298800 2,118,648,256, %e A298800 2,198,1656,2240, ... %e A298800 ... %K A298800 nonn,more,tabl,new %O A298800 0,1 %A A298800 _N. J. A. Sloane_, Feb 02 2018 %I A299038 %S A299038 1,1,1,1,1,0,1,1,1,0,1,1,1,1,0,1,1,1,2,1,0,1,1,1,2,3,1,0,1,1,1,2,4,6, %T A299038 1,0,1,1,1,2,4,8,11,1,0,1,1,1,2,4,9,17,23,1,0,1,1,1,2,4,9,19,39,46,1, %U A299038 0,1,1,1,2,4,9,20,45,89,98,1,0,1,1,1,2,4,9,20,47,106,211,207,1,0 %N A299038 Number A(n,k) of rooted trees with n nodes where each node has at most k children; square array A(n,k), n>=0, k>=0, read by antidiagonals. %H A299038 Alois P. Heinz, Antidiagonals n = 0..140, flattened %F A299038 A(n,k) = Sum_{i=0..k} A244372(n,i) for n>0, A(0,k) = 1. %e A299038 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... %e A299038 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... %e A299038 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... %e A299038 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, ... %e A299038 0, 1, 3, 4, 4, 4, 4, 4, 4, 4, 4, ... %e A299038 0, 1, 6, 8, 9, 9, 9, 9, 9, 9, 9, ... %e A299038 0, 1, 11, 17, 19, 20, 20, 20, 20, 20, 20, ... %e A299038 0, 1, 23, 39, 45, 47, 48, 48, 48, 48, 48, ... %e A299038 0, 1, 46, 89, 106, 112, 114, 115, 115, 115, 115, ... %e A299038 0, 1, 98, 211, 260, 277, 283, 285, 286, 286, 286, ... %e A299038 0, 1, 207, 507, 643, 693, 710, 716, 718, 719, 719, ... %p A299038 b:= proc(n, i, t, k) option remember; `if`(n=0, 1, %p A299038 `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)* %p A299038 b(n-i*j, i-1, t-j, k), j=0..min(t, n/i)))) %p A299038 end: %p A299038 A:= (n, k)-> `if`(n=0, 1, b(n-1$2, k$2)): %p A299038 seq(seq(A(n, d-n), n=0..d), d=0..14); %Y A299038 Columns k=1-11 give: A000012, A001190(n+1), A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556. %Y A299038 Main diagonal gives A000081 for n>0. %Y A299038 A(2n,n) gives A299039. %Y A299038 Cf. A244372. %K A299038 nonn,tabl,new %O A299038 0,19 %A A299038 _Alois P. Heinz_, Feb 01 2018 %I A299039 %S A299039 1,1,3,17,106,693,4690,32754,234746,1719325,12820920,97039824, %T A299039 743680508,5759507657,45006692668,354425763797,2809931206626, %U A299039 22409524536076,179655903886571,1447023307374888,11703779855021636,95020085240320710,774088021528328920 %N A299039 Number of rooted trees with 2n nodes where each node has at most n children. %H A299039 Alois P. Heinz, Table of n, a(n) for n = 0..275 %F A299039 a(n) = A299038(2n,n). %F A299039 a(n) ~ c * d^n / n^(3/2), where d = A051491^2 = 8.736548423865419449938118272879... and c = 0.155536626247883986039760097126... - _Vaclav Kotesovec_, Feb 02 2018 %e A299039 a(2) = 3: %e A299039 o o o %e A299039 | | / \ %e A299039 o o o o %e A299039 | / \ | %e A299039 o o o o %e A299039 | %e A299039 o %p A299039 b:= proc(n, i, t, k) option remember; `if`(n=0, 1, %p A299039 `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)* %p A299039 b(n-i*j, i-1, t-j, k), j=0..min(t, n/i)))) %p A299039 end: %p A299039 a:= n-> `if`(n=0, 1, b(2*n-1$2, n$2)): %p A299039 seq(a(n), n=0..25); %Y A299039 Cf. A051491, A100034, A244407, A299038, A299098. %K A299039 nonn,new %O A299039 0,3 %A A299039 _Alois P. Heinz_, Feb 01 2018 %I A299074 %S A299074 1,33,853,20853,502789,12080901,290025541,6961116741,167069824837, %T A299074 4009693935429,96232763288389,2309586971953989,55430091245099845, %U A299074 1330322213391637317,31927733262454774597,766265599145247529797,18390374384563938483013,441368985260002510461765 %N A299074 Expansion of 1/((1-x)*(1-2*x)*(1-6*x)*(1-24*x)). %H A299074 Colin Barker, Table of n, a(n) for n = 0..700 %H A299074 Index entries for linear recurrences with constant coefficients, signature (33,-236,492,-288). %F A299074 O.g.f.: 1/((1 - x)*(1 - 2*x)(1 - 6*x)*(1 - 24*x)). %F A299074 From _Colin Barker_, Feb 02 2018: (Start) %F A299074 a(n) = (-11 + 115*2^n - 759*6^n + 1920*24^n) / 1265. %F A299074 a(n) = 33*a(n-1) - 236*a(n-2) + 492*a(n-3) - 288*a(n-4) for n>3. (End) %o A299074 (PARI) N=66; x='x+O('x^N); Vec(1/prod(k=1, 4, (1-k!*x))) %o A299074 (PARI) Vec(1/((1 - x)*(1 - 2*x)*(1 - 6*x)*(1 - 24*x)) + O(x^20)) \\ _Colin Barker_, Feb 02 2018 %Y A299074 Cf. A126646, A016200. %K A299074 nonn,easy,new %O A299074 0,2 %A A299074 _Seiichi Manyama_, Feb 02 2018 %I A298972 %S A298972 0,0,1,0,2,0,1,0,3,0,1,2,2,0,1,0,4,0,1,0,0,0,4,0,0,0,0,0,0,0,0,0,0,5, %T A298972 0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,11,0, %U A298972 0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0 %N A298972 Number of positive integers k < n such that n occurs in the Reverse-and-Add trajectory of k. %C A298972 Number of integers k < n such that n occurs in row k of A243238. %C A298972 For n > 0, a(n) = 0 iff n is a term of A067031. %C A298972 For n > 0, a(n) > 0 iff n is a term of A067030. %H A298972 Index entries for sequences related to Reverse and Add! %e A298972 For n = 22: There exist 4 positive integers k < 22 such that 22 occurs in the Reverse-and-Add trajectory of k, namely 5, 10, 11 and 20, so a(22) = 4. %t A298972 Block[{nn = 85, s}, s = Array[Union@ NestWhileList[# + IntegerReverse@ # &, #, # < nn &, 1, nn] &, nn]; Array[Count[Take[s, # - 1], #, 2] &, nn + 1, 0]] (* _Michael De Vlieger_, Feb 01 2018 *) %o A298972 (PARI) a(n) = my(i=0); for(k=1, n-1, my(x=k); while(x < n, x=x+eval(concat(Vecrev(Str(x))))); if(x==n, i++)); i %Y A298972 Cf. A056964, A067030, A067031, A243238. %K A298972 nonn,base,new %O A298972 0,5 %A A298972 _Felix Fröhlich_, Jan 30 2018 %I A298946 %S A298946 35,462,2339,4627,2378,4238,5148,1260,57635,85026,64410,100509,163716, %T A298946 171918,93876,309780,148969,444220,370712,532771,652200,938386,816466, %U A298946 907874,569300,1107298,2470810,2953692,887812,1341810,2956584,1941390,589961,6248628 %N A298946 a(n) = binomial(2*c-1, c-1) (mod c^4), where c is the n-th composite number. %C A298946 Composites c where a(n) = 1 could be called "Wolstenholme pseudoprimes". Do any such composites exist? %C A298946 A necessary condition for c to be a "Wolstenholme pseudoprime" would be that it is a term of A228562 or A267824. %H A298946 Robert Israel, Table of n, a(n) for n = 1..10000 %p A298946 R:= NULL: %p A298946 count:= 0: F:= 10; %p A298946 for n from 4 while count < 100 do %p A298946 F:= F * (4*n-2)/n; %p A298946 if not isprime(n) then %p A298946 count:= count+1; %p A298946 R:= R, F mod (n^4); %p A298946 fi %p A298946 od: %p A298946 R; # _Robert Israel_, Feb 02 2018 %t A298946 Table[Mod[Binomial[2 c - 1, c - 1], c^4], {c, Select[Range@ 50, CompositeQ]}] (* _Michael De Vlieger_, Feb 01 2018 *) %o A298946 (PARI) forcomposite(c=1, 200, print1(lift(Mod(binomial(2*c-1, c-1), c^4)), ", ")) %o A298946 (Python) %o A298946 from sympy import binomial, composite %o A298946 def A298946(n): %o A298946 c = composite(n) %o A298946 return binomial(2*c-1,c-1) % c**4 # _Chai Wah Wu_, Feb 02 2018 %Y A298946 Cf. A088164, A228562, A244214, A267824, A281302, A298944, A298945. %K A298946 nonn,new %O A298946 1,1 %A A298946 _Felix Fröhlich_, Jan 30 2018 %I A298945 %S A298945 2,5,34,21,55,89,37,160,98,293,365,150,101,433,25,665,696,709,440,994, %T A298945 883,1090,765,1241,230,1511,1355,257,805,20,1382,289,2275,1525,1414, %U A298945 821,1373,1820,685,1504,2177,720,3102,1302,1250,190,2425,2178,2832,3935 %N A298945 a(n) = F_{c-(5/c)} mod c^2, where c is the n-th composite number, F_i = A000045(i) and (5/c) is the Kronecker symbol. %C A298945 Composites c where a(n) = 0 could be called "Wall-Sun-Sun pseudoprimes" or "Fibonacci-Wieferich pseudoprimes". Do any such composites exist? %C A298945 Any such c would have to be a term of A241505. %H A298945 Robert Israel, Table of n, a(n) for n = 1..10000 %p A298945 N:= 100: # to get a(1)..a(N) %p A298945 count:= 0: R:= NULL: %p A298945 for n from 4 while count < N do %p A298945 if not isprime(n) then %p A298945 count:= count+1; %p A298945 R:= R, combinat:-fibonacci(n - numtheory:-jacobi(5,n)) mod n^2; %p A298945 fi %p A298945 od: %p A298945 R; # _Robert Israel_, Feb 02 2018 %t A298945 composite[n_Integer] := FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1] ; Array[With[{c = composite@ #}, Mod[Fibonacci[c - KroneckerSymbol[5, c]], c^2]] &, 50] (* _Michael De Vlieger_, Jan 31 2018, composite function by _Robert G. Wilson v_ at A066277 *) %o A298945 (PARI) forcomposite(c=1, 200, print1(lift(Mod(fibonacci(c-kronecker(5, c)), c^2)), ", ")) %Y A298945 Cf. A000045, A241505, A298944, A298946. %K A298945 nonn,new %O A298945 1,1 %A A298945 _Felix Fröhlich_, Jan 30 2018 %I A299097 %S A299097 1,2,2,4,7,4,8,13,13,8,16,29,20,29,16,32,73,41,41,73,32,64,157,101, %T A299097 125,101,157,64,128,353,242,574,574,242,353,128,256,869,578,1847,2828, %U A299097 1847,578,869,256,512,1993,1385,6007,9624,9624,6007,1385,1993,512,1024,4557 %N A299097 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299097 Table starts %C A299097 ...1....2....4.....8.....16......32........64........128.........256 %C A299097 ...2....7...13....29.....73.....157.......353........869........1993 %C A299097 ...4...13...20....41....101.....242.......578.......1385........3368 %C A299097 ...8...29...41...125....574....1847......6007......22330.......78424 %C A299097 ..16...73..101...574...2828....9624.....44936.....204059......865754 %C A299097 ..32..157..242..1847...9624...42012....255009....1414647.....7786685 %C A299097 ..64..353..578..6007..44936..255009...2170819...16528508...123272050 %C A299097 .128..869.1385.22330.204059.1414647..16528508..161247816..1570964262 %C A299097 .256.1993.3368.78424.865754.7786685.123272050.1570964262.20536604982 %H A299097 R. H. Hardin, Table of n, a(n) for n = 1..199 %F A299097 Empirical for column k: %F A299097 k=1: a(n) = 2*a(n-1) %F A299097 k=2: a(n) = 4*a(n-1) -5*a(n-2) +10*a(n-3) -24*a(n-4) +16*a(n-5) for n>6 %F A299097 k=3: [order 16] for n>17 %F A299097 k=4: [order 68] for n>69 %e A299097 Some solutions for n=5 k=4 %e A299097 ..0..1..0..1. .0..0..0..1. .0..1..1..0. .0..1..0..1. .0..1..0..1 %e A299097 ..1..0..1..1. .1..0..0..0. .0..0..1..0. .0..1..0..0. .0..1..0..0 %e A299097 ..0..0..1..1. .0..0..0..0. .0..0..1..1. .0..1..0..0. .1..1..1..1 %e A299097 ..1..1..1..0. .0..0..0..1. .1..0..1..1. .0..1..0..0. .0..0..1..0 %e A299097 ..1..0..1..0. .1..0..0..0. .0..1..0..1. .0..1..0..1. .1..0..1..1 %Y A299097 Column 1 is A000079(n-1). %Y A299097 Column 2 is A298215. %K A299097 nonn,tabl,new %O A299097 1,2 %A A299097 _R. H. Hardin_, Feb 02 2018 %I A299096 %S A299096 64,353,578,6007,44936,255009,2170819,16528508,123272050,976854712, %T A299096 7530554085,58454112878,457202830027,3549779070263,27660647795555, %U A299096 215783822807485,1679905974605338,13091806853510818,102058138906901208 %N A299096 Number of nX7 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299096 Column 7 of A299097. %H A299096 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299096 Some solutions for n=5 %e A299096 ..0..1..0..1..1..1..0. .0..1..1..0..1..1..1. .0..1..0..1..1..1..1 %e A299096 ..0..0..1..0..0..0..1. .0..1..0..0..1..0..1. .1..1..0..0..0..0..0 %e A299096 ..1..1..0..1..0..0..0. .0..1..1..1..1..1..1. .0..0..0..1..0..0..1 %e A299096 ..0..1..1..1..1..1..1. .0..1..1..0..1..1..1. .1..1..0..0..0..0..0 %e A299096 ..1..0..0..0..1..1..0. .0..1..0..0..1..0..1. .0..1..0..1..1..1..1 %Y A299096 Cf. A299097. %K A299096 nonn,new %O A299096 1,1 %A A299096 _R. H. Hardin_, Feb 02 2018 %I A299095 %S A299095 32,157,242,1847,9624,42012,255009,1414647,7786685,44969792,255356420, %T A299095 1455035527,8350312521,47760326861,273535579360,1568756314490, %U A299095 8990344477500,51538983266382,295559066895418,1694685265216804 %N A299095 Number of nX6 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299095 Column 6 of A299097. %H A299095 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299095 Some solutions for n=5 %e A299095 ..0..0..1..0..0..1. .0..0..1..0..1..1. .0..1..0..1..0..1. .0..1..1..0..1..1 %e A299095 ..1..0..1..1..0..1. .1..0..1..0..1..0. .0..0..1..0..0..1. .0..0..1..0..0..0 %e A299095 ..1..1..1..0..1..0. .1..1..1..1..0..1. .0..0..1..1..0..0. .0..0..0..0..1..1 %e A299095 ..0..0..1..1..0..1. .1..1..1..0..1..0. .1..0..1..1..0..0. .0..0..0..0..1..0 %e A299095 ..1..0..1..0..0..1. .1..0..1..0..1..1. .0..1..0..0..1..0. .1..0..0..1..0..1 %Y A299095 Cf. A299097. %K A299095 nonn,new %O A299095 1,1 %A A299095 _R. H. Hardin_, Feb 02 2018 %I A299094 %S A299094 16,73,101,574,2828,9624,44936,204059,865754,3858423,17203848, %T A299094 76120565,337435515,1499132398,6667134660,29596501785,131442334320, %U A299094 584408632215,2596483900963,11534083214519,51261067853186,227794028537569 %N A299094 Number of nX5 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299094 Column 5 of A299097. %H A299094 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299094 Some solutions for n=5 %e A299094 ..0..1..0..0..0. .0..1..0..1..0. .0..0..1..0..1. .0..1..0..1..1 %e A299094 ..1..1..0..1..0. .1..0..0..1..1. .1..1..0..1..0. .1..1..0..0..0 %e A299094 ..0..0..0..0..0. .1..0..0..0..0. .0..1..1..1..0. .0..0..0..1..1 %e A299094 ..1..0..1..1..1. .1..0..0..0..1. .0..1..1..1..1. .1..1..0..0..0 %e A299094 ..0..1..0..0..0. .0..1..0..0..0. .1..0..0..0..0. .0..0..1..1..0 %Y A299094 Cf. A299097. %K A299094 nonn,new %O A299094 1,1 %A A299094 _R. H. Hardin_, Feb 02 2018 %I A299093 %S A299093 8,29,41,125,574,1847,6007,22330,78424,268599,949084,3349021,11698218, %T A299093 41025548,144230504,506030772,1774857783,6230482209,21869669571, %U A299093 76741039322,269319672263,945257932761,3317391481932,11642270682192 %N A299093 Number of nX4 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299093 Column 4 of A299097. %H A299093 R. H. Hardin, Table of n, a(n) for n = 1..210 %H A299093 R. H. Hardin, Empirical recurrence of order 68 %F A299093 Empirical recurrence of order 68 (see link above) %e A299093 Some solutions for n=5 %e A299093 ..0..1..1..0. .0..0..1..0. .0..0..1..1. .0..1..1..1. .0..1..0..0 %e A299093 ..1..1..1..1. .1..1..1..0. .1..0..1..0. .0..1..0..1. .0..1..1..1 %e A299093 ..1..1..1..1. .1..0..1..1. .0..0..0..0. .1..1..1..1. .1..1..1..0 %e A299093 ..0..1..1..0. .1..1..0..0. .0..0..1..1. .1..1..0..0. .0..0..0..1 %e A299093 ..0..0..1..0. .0..0..1..0. .1..0..1..0. .0..1..0..1. .1..0..0..0 %Y A299093 Cf. A299097. %K A299093 nonn,new %O A299093 1,1 %A A299093 _R. H. Hardin_, Feb 02 2018 %I A299092 %S A299092 4,13,20,41,101,242,578,1385,3368,8216,20014,48885,119555,292427, %T A299092 715827,1753080,4294103,10521578,25786055,63204089,154942532, %U A299092 379879302,931443529,2284029958,5601114152,13736265904,33688529855,82624915345 %N A299092 Number of nX3 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299092 Column 3 of A299097. %H A299092 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299092 Empirical: a(n) = 4*a(n-1) -3*a(n-2) +4*a(n-3) -23*a(n-4) +18*a(n-5) -8*a(n-6) +47*a(n-7) -29*a(n-8) +26*a(n-9) -66*a(n-10) +17*a(n-11) -29*a(n-12) +46*a(n-13) -3*a(n-14) +18*a(n-15) -18*a(n-16) for n>17 %e A299092 Some solutions for n=5 %e A299092 ..0..1..0. .0..1..0. .0..1..0. .0..1..0. .0..1..0. .0..0..0. .0..1..0 %e A299092 ..0..0..1. .0..1..1. .1..0..1. .0..0..0. .0..0..0. .1..1..1. .0..0..0 %e A299092 ..0..0..1. .0..1..1. .1..0..0. .0..0..0. .1..1..1. .0..0..0. .1..1..1 %e A299092 ..1..0..0. .0..1..0. .1..0..0. .0..0..1. .0..0..0. .1..1..1. .1..0..1 %e A299092 ..0..1..1. .1..1..0. .0..1..0. .1..0..1. .1..1..1. .0..0..0. .1..1..1 %Y A299092 Cf. A299097. %K A299092 nonn,new %O A299092 1,1 %A A299092 _R. H. Hardin_, Feb 02 2018 %I A299091 %S A299091 1,7,20,125,2828,42012,2170819,161247816,20536604982,5567500327979 %N A299091 Number of nXn 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299091 Diagonal of A299097. %e A299091 Some solutions for n=5 %e A299091 ..0..1..0..0..0. .0..1..0..1..0. .0..1..0..1..1. .0..1..0..1..0 %e A299091 ..1..0..1..1..1. .1..0..1..0..1. .0..0..1..0..0. .1..1..0..0..1 %e A299091 ..0..0..0..0..0. .0..1..0..1..0. .0..0..1..1..1. .0..0..0..0..1 %e A299091 ..1..1..0..1..0. .0..1..1..1..0. .1..0..1..1..0. .1..0..0..0..1 %e A299091 ..0..1..0..1..1. .1..1..1..1..1. .0..1..0..0..1. .1..0..1..0..1 %Y A299091 Cf. A299097. %K A299091 nonn,new %O A299091 1,2 %A A299091 _R. H. Hardin_, Feb 02 2018 %I A299089 %S A299089 1,2,2,4,8,4,8,26,26,8,16,88,92,88,16,32,298,354,354,298,32,64,1012, %T A299089 1387,1617,1387,1012,64,128,3440,5470,7722,7722,5470,3440,128,256, %U A299089 11700,21484,36667,46456,36667,21484,11700,256,512,39804,84425,173524,273360,273360 %N A299089 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299089 Table starts %C A299089 ...1.....2......4.......8.......16........32..........64..........128 %C A299089 ...2.....8.....26......88......298......1012........3440........11700 %C A299089 ...4....26.....92.....354.....1387......5470.......21484........84425 %C A299089 ...8....88....354....1617.....7722.....36667......173524.......822065 %C A299089 ..16...298...1387....7722....46456....273360.....1598956......9400094 %C A299089 ..32..1012...5470...36667...273360...2001653....14514670....105596164 %C A299089 ..64..3440..21484..173524..1598956..14514670...129999453...1166134994 %C A299089 .128.11700..84425..822065..9400094.105596164..1166134994..12899638332 %C A299089 .256.39804.331838.3897261.55296169.769538056.10493701031.143316205219 %H A299089 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299089 Empirical for column k: %F A299089 k=1: a(n) = 2*a(n-1) %F A299089 k=2: a(n) = 4*a(n-1) -2*a(n-2) +2*a(n-3) -6*a(n-4) -4*a(n-5) %F A299089 k=3: [order 17] for n>19 %F A299089 k=4: [order 64] for n>66 %e A299089 Some solutions for n=5 k=4 %e A299089 ..0..0..1..1. .0..0..0..1. .0..0..0..1. .0..0..1..1. .0..1..1..1 %e A299089 ..1..0..0..0. .0..1..1..1. .1..1..1..0. .1..1..1..0. .1..0..0..0 %e A299089 ..0..1..0..0. .0..1..1..0. .1..1..1..1. .1..1..0..1. .0..0..0..0 %e A299089 ..1..0..0..0. .0..1..1..1. .1..1..0..0. .1..1..1..1. .1..1..0..0 %e A299089 ..0..0..1..1. .0..0..0..1. .0..0..1..0. .0..0..1..0. .1..0..1..1 %Y A299089 Column 1 is A000079(n-1). %Y A299089 Column 2 is A298189. %K A299089 nonn,tabl,new %O A299089 1,2 %A A299089 _R. H. Hardin_, Feb 02 2018 %I A299088 %S A299088 64,3440,21484,173524,1598956,14514670,129999453,1166134994, %T A299088 10493701031,94438101000,849751415904,7648233297772,68841581935753, %U A299088 619625217826871,5577260749794146,50202038501971691,451878831715700135 %N A299088 Number of nX7 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299088 Column 7 of A299089. %H A299088 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299088 Some solutions for n=5 %e A299088 ..0..1..0..0..0..1..1. .0..1..1..1..1..0..1. .0..1..1..1..0..0..1 %e A299088 ..0..1..1..1..1..0..0. .0..1..0..0..0..0..0. .0..0..0..1..1..0..1 %e A299088 ..0..1..0..1..0..0..0. .0..1..0..0..0..1..0. .0..1..0..1..1..1..0 %e A299088 ..0..1..0..0..0..0..0. .0..1..0..0..1..1..0. .0..1..0..1..1..1..0 %e A299088 ..0..1..1..0..1..1..1. .0..0..1..0..1..0..0. .0..0..1..1..0..1..1 %Y A299088 Cf. A299089. %K A299088 nonn,new %O A299088 1,1 %A A299088 _R. H. Hardin_, Feb 02 2018 %I A299087 %S A299087 32,1012,5470,36667,273360,2001653,14514670,105596164,769538056, %T A299087 5605624825,40835714546,297572901733,2168362433721,15799872853544, %U A299087 115131275258487,838953244505743,6113360789406252,44547557450851112 %N A299087 Number of nX6 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299087 Column 6 of A299089. %H A299087 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299087 Some solutions for n=5 %e A299087 ..0..1..1..1..0..1. .0..1..0..1..0..0. .0..1..1..0..1..0. .0..1..1..0..0..1 %e A299087 ..1..0..0..0..0..1. .1..1..1..1..1..1. .1..0..0..0..0..0. .1..0..1..1..1..0 %e A299087 ..0..1..1..0..0..0. .0..1..1..1..0..1. .0..1..1..1..1..0. .0..1..1..0..1..0 %e A299087 ..0..0..1..0..1..1. .0..1..0..1..0..0. .0..0..0..1..0..1. .0..1..1..0..1..0 %e A299087 ..1..1..1..1..0..0. .1..0..0..1..0..1. .1..1..1..1..0..0. .1..0..1..1..0..0 %Y A299087 Cf. A299089. %K A299087 nonn,new %O A299087 1,1 %A A299087 _R. H. Hardin_, Feb 02 2018 %I A299086 %S A299086 16,298,1387,7722,46456,273360,1598956,9400094,55296169,324970247, %T A299086 1910282856,11232387554,66037705308,388243985602,2282640753279, %U A299086 13420442718371,78902979338310,463898209859700,2727419452854293 %N A299086 Number of nX5 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299086 Column 5 of A299089. %H A299086 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299086 Some solutions for n=5 %e A299086 ..0..1..0..0..0. .0..0..1..0..1. .0..1..0..1..0. .0..1..1..1..0 %e A299086 ..1..1..0..0..0. .1..0..0..0..1. .0..1..1..1..1. .0..1..1..1..0 %e A299086 ..0..0..1..0..0. .1..0..0..0..0. .1..1..1..1..0. .1..0..1..1..0 %e A299086 ..0..0..0..0..0. .1..1..1..1..0. .0..1..0..1..0. .0..1..1..1..0 %e A299086 ..0..0..0..0..0. .0..0..0..1..1. .1..0..1..1..1. .1..0..1..0..1 %Y A299086 Cf. A299089. %K A299086 nonn,new %O A299086 1,1 %A A299086 _R. H. Hardin_, Feb 02 2018 %I A299085 %S A299085 8,88,354,1617,7722,36667,173524,822065,3897261,18474589,87565905, %T A299085 415072759,1967506932,9326121377,44206712718,209544659717, %U A299085 993262158136,4708160655145,22317159667937,105785574659767,501434203894082 %N A299085 Number of nX4 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299085 Column 4 of A299089. %H A299085 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299085 Empirical: a(n) = 8*a(n-1) -19*a(n-2) +43*a(n-3) -177*a(n-4) +250*a(n-5) -111*a(n-6) +598*a(n-7) +175*a(n-8) -2033*a(n-9) +835*a(n-10) -6610*a(n-11) +4604*a(n-12) +5357*a(n-13) +10181*a(n-14) +13348*a(n-15) -21106*a(n-16) -6122*a(n-17) -36475*a(n-18) +37696*a(n-19) +65360*a(n-20) +163618*a(n-21) -534123*a(n-22) +107096*a(n-23) -1235788*a(n-24) +1583118*a(n-25) +112658*a(n-26) +2520965*a(n-27) -639433*a(n-28) -3386070*a(n-29) -890603*a(n-30) -4627333*a(n-31) +6206726*a(n-32) +2036692*a(n-33) +6154747*a(n-34) +1833433*a(n-35) -4816911*a(n-36) -1469263*a(n-37) -8046242*a(n-38) +544975*a(n-39) -251241*a(n-40) +2025611*a(n-41) +2899322*a(n-42) -384613*a(n-43) +1524228*a(n-44) -1083347*a(n-45) -62502*a(n-46) -198019*a(n-47) -547484*a(n-48) +288668*a(n-49) -24160*a(n-50) +124473*a(n-51) +63988*a(n-52) -63036*a(n-53) +37665*a(n-54) -27918*a(n-55) -14368*a(n-56) +9942*a(n-57) -1736*a(n-58) -646*a(n-59) +974*a(n-60) -64*a(n-61) +264*a(n-62) -64*a(n-64) for n>66 %e A299085 Some solutions for n=5 %e A299085 ..0..0..1..0. .0..0..1..0. .0..1..1..0. .0..0..1..0. .0..1..1..0 %e A299085 ..1..1..1..1. .1..1..1..1. .0..0..0..0. .1..1..1..1. .0..0..0..0 %e A299085 ..1..1..0..1. .1..1..0..1. .1..0..0..1. .0..1..1..0. .1..0..0..1 %e A299085 ..1..1..1..0. .1..1..1..0. .0..0..0..0. .1..1..1..1. .0..0..0..0 %e A299085 ..0..0..1..1. .1..1..1..0. .1..1..0..1. .1..0..0..1. .1..0..1..1 %Y A299085 Cf. A299089. %K A299085 nonn,new %O A299085 1,1 %A A299085 _R. H. Hardin_, Feb 02 2018 %I A299084 %S A299084 4,26,92,354,1387,5470,21484,84425,331838,1304618,5128566,20161270, %T A299084 79257643,311578924,1224880378,4815257718,18929769479,74416841347, %U A299084 292547978257,1150066574420,4521149390460,17773572810348,69871588449166 %N A299084 Number of nX3 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299084 Column 3 of A299089. %H A299084 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299084 Empirical: a(n) = 5*a(n-1) -4*a(n-2) -2*a(n-3) +9*a(n-4) -23*a(n-5) +30*a(n-6) -23*a(n-7) +24*a(n-8) -64*a(n-9) -12*a(n-10) +66*a(n-11) +38*a(n-12) -37*a(n-13) -44*a(n-14) +4*a(n-15) +22*a(n-16) +8*a(n-17) for n>19 %e A299084 Some solutions for n=6 %e A299084 ..0..0..1. .0..0..1. .0..1..1. .0..1..1. .0..0..1. .0..1..1. .0..0..1 %e A299084 ..1..0..0. .1..1..0. .0..0..0. .0..0..0. .1..1..1. .0..0..0. .1..1..1 %e A299084 ..1..1..1. .0..0..0. .1..0..0. .1..0..0. .1..1..0. .1..0..0. .1..1..0 %e A299084 ..0..1..1. .0..0..1. .0..0..0. .0..0..0. .1..1..1. .0..0..0. .1..1..1 %e A299084 ..1..1..1. .0..0..0. .0..1..1. .0..1..1. .0..0..1. .0..1..1. .0..0..1 %e A299084 ..1..0..0. .1..1..0. .0..1..1. .1..0..1. .1..0..0. .1..1..0. .0..1..0 %Y A299084 Cf. A299089. %K A299084 nonn,new %O A299084 1,1 %A A299084 _R. H. Hardin_, Feb 02 2018 %I A299083 %S A299083 1,8,92,1617,46456,2001653,129999453,12899638332,1967580379354 %N A299083 Number of nXn 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 7 king-move adjacent elements, with upper left element zero. %C A299083 Diagonal of A299089. %e A299083 Some solutions for n=5 %e A299083 ..0..1..0..1..0. .0..1..1..1..1. .0..1..0..0..0. .0..0..0..0..0 %e A299083 ..0..0..1..0..1. .0..0..0..0..1. .1..1..1..1..0. .1..1..1..1..0 %e A299083 ..1..0..0..0..0. .1..0..0..0..1. .0..1..1..0..1. .1..0..0..0..1 %e A299083 ..1..0..0..0..1. .1..0..1..0..0. .1..1..1..1..0. .1..0..0..0..0 %e A299083 ..0..0..1..0..1. .0..1..1..1..1. .0..1..0..0..0. .0..0..1..0..1 %Y A299083 Cf. A299089. %K A299083 nonn,new %O A299083 1,2 %A A299083 _R. H. Hardin_, Feb 02 2018 %I A299081 %S A299081 1,2,2,4,8,4,8,31,31,8,16,121,163,121,16,32,472,927,927,472,32,64, %T A299081 1841,5331,8245,5331,1841,64,128,7181,30535,74329,74329,30535,7181, %U A299081 128,256,28010,175286,664377,1055999,664377,175286,28010,256,512,109255,1006611 %N A299081 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 7 king-move adjacent elements, with upper left element zero. %C A299081 Table starts %C A299081 ...1......2.......4.........8..........16............32..............64 %C A299081 ...2......8......31.......121.........472..........1841............7181 %C A299081 ...4.....31.....163.......927........5331.........30535..........175286 %C A299081 ...8....121.....927......8245.......74329........664377.........5966905 %C A299081 ..16....472....5331.....74329.....1055999......14831527.......209649724 %C A299081 ..32...1841...30535....664377....14831527.....325878090......7216923643 %C A299081 ..64...7181..175286...5966905...209649724....7216923643....250915355467 %C A299081 .128..28010.1006611..53667656..2971224409..160460154889...8769918141746 %C A299081 .256.109255.5780036.482603686.42096980790.3565905309360.306309018507458 %H A299081 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299081 Empirical for column k: %F A299081 k=1: a(n) = 2*a(n-1) %F A299081 k=2: a(n) = 3*a(n-1) +3*a(n-2) +2*a(n-3) %F A299081 k=3: [order 10] for n>11 %F A299081 k=4: [order 30] for n>31 %e A299081 Some solutions for n=5 k=4 %e A299081 ..0..1..0..1. .0..1..0..1. .0..1..1..1. .0..0..0..0. .0..1..1..0 %e A299081 ..0..1..1..1. .1..1..1..1. .0..0..0..0. .1..0..0..1. .1..1..1..1 %e A299081 ..0..1..1..1. .0..0..0..0. .1..1..1..1. .0..0..0..0. .1..1..1..1 %e A299081 ..0..1..0..1. .1..0..0..1. .0..1..1..0. .1..1..1..1. .0..1..1..0 %e A299081 ..0..0..0..1. .0..0..0..0. .1..1..1..1. .1..1..0..1. .1..1..1..1 %Y A299081 Column 1 is A000079(n-1). %Y A299081 Column 2 is A281831. %K A299081 nonn,tabl,new %O A299081 1,2 %A A299081 _R. H. Hardin_, Feb 02 2018 %I A299080 %S A299080 64,7181,175286,5966905,209649724,7216923643,250915355467, %T A299080 8769918141746,306309018507458,10700121813906426,373936136661714476, %U A299080 13068516356658656411,456719559953077301270,15961823564161742473972 %N A299080 Number of nX7 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 7 king-move adjacent elements, with upper left element zero. %C A299080 Column 7 of A299081. %H A299080 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299080 Some solutions for n=3 %e A299080 ..0..0..0..1..1..1..1. .0..0..0..0..0..1..1. .0..0..1..1..1..0..1 %e A299080 ..0..0..1..0..1..1..0. .1..1..1..1..1..0..1. .1..1..1..1..0..1..1 %e A299080 ..1..1..1..1..1..1..1. .0..1..1..0..0..1..0. .0..0..0..0..0..0..1 %Y A299080 Cf. A299081. %K A299080 nonn,new %O A299080 1,1 %A A299080 _R. H. Hardin_, Feb 02 2018 %I A299079 %S A299079 32,1841,30535,664377,14831527,325878090,7216923643,160460154889, %T A299079 3565905309360,79255889463567,1762053804143828,39176124976515380, %U A299079 871006563676257796,19365455206372348252,430563149744320860362 %N A299079 Number of nX6 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 7 king-move adjacent elements, with upper left element zero. %C A299079 Column 6 of A299081. %H A299079 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299079 Some solutions for n=3 %e A299079 ..0..1..1..1..1..1. .0..1..0..0..0..0. .0..1..1..1..1..1. .0..1..1..1..1..1 %e A299079 ..0..1..0..1..1..0. .1..0..1..0..0..1. .1..0..1..1..0..0. .0..0..1..1..0..0 %e A299079 ..0..1..1..1..1..1. .0..1..0..0..0..0. .1..1..1..1..1..0. .0..1..1..1..1..0 %Y A299079 Cf. A299081. %K A299079 nonn,new %O A299079 1,1 %A A299079 _R. H. Hardin_, Feb 02 2018 %I A299078 %S A299078 16,472,5331,74329,1055999,14831527,209649724,2971224409,42096980790, %T A299078 596526427163,8454487189243,119826774222264,1698327003054369, %U A299078 24070925568501537,341165960650230205,4835470420420172404 %N A299078 Number of nX5 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 7 king-move adjacent elements, with upper left element zero. %C A299078 Column 5 of A299081. %H A299078 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299078 Some solutions for n=5 %e A299078 ..0..0..0..1..0. .0..0..0..0..1. .0..0..0..0..0. .0..0..0..1..0 %e A299078 ..0..1..1..1..0. .1..0..0..1..0. .0..1..0..0..1. .0..1..1..1..0 %e A299078 ..0..0..1..0..0. .0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0 %e A299078 ..1..0..1..0..1. .1..1..1..1..1. .1..1..1..1..1. .0..1..1..1..0 %e A299078 ..1..0..1..0..1. .1..0..0..0..0. .0..0..0..0..1. .0..1..0..0..0 %Y A299078 Cf. A299081. %K A299078 nonn,new %O A299078 1,1 %A A299078 _R. H. Hardin_, Feb 02 2018 %I A299077 %S A299077 8,121,927,8245,74329,664377,5966905,53667656,482603686,4340430649, %T A299077 39040203717,351150199077,3158465607310,28409347903144, %U A299077 255532834427292,2298435315054163,20673687740237324,185953195378108520 %N A299077 Number of nX4 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 7 king-move adjacent elements, with upper left element zero. %C A299077 Column 4 of A299081. %H A299077 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299077 Empirical: a(n) = 9*a(n-1) +37*a(n-3) -292*a(n-4) -375*a(n-5) -491*a(n-6) +1814*a(n-7) +3581*a(n-8) +3193*a(n-9) -3202*a(n-10) -11261*a(n-11) -6211*a(n-12) -1851*a(n-13) +14158*a(n-14) -1752*a(n-15) +10727*a(n-16) -10489*a(n-17) +10142*a(n-18) -13135*a(n-19) +8571*a(n-20) -4435*a(n-21) +6805*a(n-22) -3831*a(n-23) +947*a(n-24) -578*a(n-25) +842*a(n-26) -308*a(n-27) +40*a(n-28) -154*a(n-29) +40*a(n-30) for n>31 %e A299077 Some solutions for n=5 %e A299077 ..0..0..0..0. .0..1..0..1. .0..1..0..1. .0..1..0..1. .0..1..0..1 %e A299077 ..1..0..0..1. .0..1..1..1. .0..1..1..1. .0..0..0..0. .1..0..1..0 %e A299077 ..0..0..0..0. .0..1..1..1. .0..1..1..1. .1..1..1..1. .1..0..0..0 %e A299077 ..0..0..0..0. .0..1..0..1. .0..1..0..1. .0..1..1..0. .1..0..0..0 %e A299077 ..1..0..0..1. .0..0..1..1. .1..0..0..1. .1..1..1..1. .1..0..1..0 %Y A299077 Cf. A299081. %K A299077 nonn,new %O A299077 1,1 %A A299077 _R. H. Hardin_, Feb 02 2018 %I A299076 %S A299076 4,31,163,927,5331,30535,175286,1006611,5780036,33191586,190603429, %T A299076 1094541845,6285428287,36094210900,207271793585,1190262866608, %U A299076 6835110967356,39250776619893,225398457372461,1294355653107337 %N A299076 Number of nX3 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 7 king-move adjacent elements, with upper left element zero. %C A299076 Column 3 of A299081. %H A299076 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299076 Empirical: a(n) = 5*a(n-1) +4*a(n-2) +7*a(n-3) -26*a(n-4) -28*a(n-5) -18*a(n-6) -16*a(n-7) +12*a(n-8) +8*a(n-9) +16*a(n-10) for n>11 %e A299076 Some solutions for n=5 %e A299076 ..0..1..0. .0..0..1. .0..1..0. .0..1..0. .0..1..0. .0..1..0. .0..0..0 %e A299076 ..0..0..0. .1..0..1. .0..0..0. .0..0..0. .0..0..0. .0..0..0. .1..0..1 %e A299076 ..0..0..0. .1..1..1. .0..0..0. .0..0..0. .0..0..0. .0..0..0. .1..1..1 %e A299076 ..0..1..0. .1..1..1. .0..1..0. .0..1..0. .0..1..0. .0..1..0. .1..1..1 %e A299076 ..0..1..1. .1..0..1. .1..0..0. .0..0..1. .0..0..0. .1..0..1. .1..0..1 %Y A299076 Cf. A299081. %K A299076 nonn,new %O A299076 1,1 %A A299076 _R. H. Hardin_, Feb 02 2018 %I A299075 %S A299075 1,8,163,8245,1055999,325878090,250915355467,482492338693115, %T A299075 2293605474526180949 %N A299075 Number of nXn 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 7 king-move adjacent elements, with upper left element zero. %C A299075 Diagonal of A299081. %e A299075 Some solutions for n=4 %e A299075 ..0..1..0..1. .0..1..1..0. .0..0..0..0. .0..1..0..1. .0..0..0..0 %e A299075 ..0..1..1..1. .1..1..1..1. .1..1..1..1. .0..0..0..1. .1..0..0..1 %e A299075 ..0..1..1..1. .1..1..1..1. .0..1..1..0. .0..0..0..1. .0..0..0..0 %e A299075 ..0..1..0..1. .0..1..1..0. .1..1..1..1. .0..1..0..1. .1..1..1..1 %Y A299075 Cf. A299081. %K A299075 nonn,new %O A299075 1,2 %A A299075 _R. H. Hardin_, Feb 02 2018 %I A298851 %S A298851 1,1,21,1408,196053,46587905,16875270660,8657594647800, %T A298851 5974284925007685,5336898188553325075,5992171630749371157181, %U A298851 8260051854943114812198756,13714895317396748230146099660,26998129079190909699998105620908 %N A298851 a(n) = [x^n] Product_{k=1..n} 1/(1-k^2*x). %H A298851 Vaclav Kotesovec, Table of n, a(n) for n = 0..200 %F A298851 a(n) ~ c * d^n * n^(2*n - 1/2), where d = 1.774513671664430848697327843228386312953174421074432567764556466987... and c = 0.617929515483613293691991371141292259390065108300160936187723552669... - _Vaclav Kotesovec_, Feb 02 2018 %t A298851 Table[SeriesCoefficient[Product[1/(1 - k^2*x), {k, 1, n}], {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Feb 02 2018 *) %Y A298851 Cf. A001044. %Y A298851 Cf. A007820, A299035, A299036. %K A298851 nonn,new %O A298851 0,3 %A A298851 _Seiichi Manyama_, Feb 01 2018 %I A299035 %S A299035 1,1,21,23980,4896624249,327969374429859111, %T A299035 11123496833223144303532943536, %U A299035 273486179312859032380857823231575174373792,6620886635410516590847876477644821623913997428738363459941 %N A299035 a(n) = [x^n] Product_{k=1..n} 1/(1-k^k*x). %H A299035 Seiichi Manyama, Table of n, a(n) for n = 0..26 %F A299035 a(n) ~ n^(n^2). - _Vaclav Kotesovec_, Feb 02 2018 %t A299035 Table[SeriesCoefficient[Product[1/(1 - k^k*x), {k, 1, n}], {x, 0, n}], {n, 0, 10}] (* _Vaclav Kotesovec_, Feb 02 2018 *) %Y A299035 Cf. A002109. %Y A299035 Cf. A007820, A298851, A299036. %K A299035 nonn,new %O A299035 0,3 %A A299035 _Seiichi Manyama_, Feb 01 2018 %I A299036 %S A299036 1,1,7,381,502789,33572762781,175123095782787181, %T A299036 99374457734129265819664221,8158897372191288496224413025490409437, %U A299036 124778468912108975502836576328262294089846582756189 %N A299036 a(n) = [x^n] Product_{k=1..n} 1/(1-k!*x). %H A299036 Seiichi Manyama, Table of n, a(n) for n = 0..30 %F A299036 From _Vaclav Kotesovec_, Feb 02 2018: (Start) %F A299036 a(n) ~ (n!)^n. %F A299036 a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2 - 1/12). (End) %t A299036 Table[SeriesCoefficient[Product[1/(1 - k!*x), {k, 1, n}], {x, 0, n}], {n, 0, 12}] (* _Vaclav Kotesovec_, Feb 02 2018 *) %Y A299036 Cf. A000178, A036740. %Y A299036 Cf. A007820, A298851, A299035. %K A299036 nonn,new %O A299036 0,3 %A A299036 _Seiichi Manyama_, Feb 01 2018 %I A299017 %S A299017 1,3,6,10,21,36,55,78,105,136,171,210,253,300,351 %N A299017 Intersection of A264041 and A000217. %F A299017 Conjectures (Start) %F A299017 G.f.: x*(1 + 6*x^4 - 3*x^5) / (1 - x)^3. %F A299017 a(n) = 6 - 7*n + 2*n^2 for n>3. %F A299017 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. %F A299017 (End) %Y A299017 Cf. A000217, A264041. %K A299017 nonn,more,new %O A299017 1,2 %A A299017 _Colin Barker_, Jan 31 2018 %I A298984 %S A298984 1,3,7,8,9,13,14,22,30,33,34,43,49,51,55,56,62,66,73,76,83,90,91,92, %T A298984 94,95,96,98,99,103,109,113,127,129,130,132,133,137,139,141,150,154, %U A298984 159,169,170,174,175,177,179,180,181,185,186,192,194,202,208,211,215 %N A298984 Numbers k such that floor((10^p) / k) has digital sum k for some integer p. %C A298984 This sequence has similarities with A106039: here some partial sum of digits of 1/k equals k, there some partial (cyclical) sum of digits of k equals k. %C A298984 A052268 is a subsequence. %e A298984 floor(1000 / 7) = 142 and 1 + 4 + 2 = 7, hence 7 belongs to this sequence. %e A298984 floor(1 / 5) = 0 and floor ((10^p) / 5) = 2 for any p > 0, hence 5 does not belong to this sequence. %o A298984 (PARI) is(n) = my (r=1/n, s=0); while (r, s+=floor(r); if (s==n, return (1), s>n, return (0); r = frac(r)*10); return (0) %Y A298984 Cf. A007953, A052268, A106039. %K A298984 nonn,base,easy,new %O A298984 1,2 %A A298984 _Rémy Sigrist_, Jan 31 2018 %I A299034 %S A299034 1,1,8,93,1544,32615,843264,25739539,906373376,36163950849, %T A299034 1612483625600,79458277381901,4288069172500992,251520785449249927, %U A299034 15932801526165085184,1084003570689331039875,78835487923639854792704,6103175938145968656408641,501114006272655771562911744 %N A299034 a(n) = n! * [x^n] Product_{k>=1} 1/(1 - x^k)^(n/k). %F A299034 a(n) = n! * [x^n] exp(n*Sum_{k>=1} d(k)*x^k/k), where d(k) is the number of divisors of k (A000005). %e A299034 The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(n/k) begins: %e A299034 n = 0: (1), 0, 0, 0, 0, 0, 0, ... %e A299034 n = 1: 1, (1), 3, 11, 59, 339, 2629, ... %e A299034 n = 2: 1, 2, (8), 40, 260, 1928, 17056, ... %e A299034 n = 3: 1, 3, 15, (93), 711, 6237, 62901, ... %e A299034 n = 4: 1, 4, 24, 176, (1544), 15456, 174784, ... %e A299034 n = 5: 1, 5, 35, 295, 2915, (32615), 407725, ... %e A299034 n = 6: 1, 6, 48, 456, 5004, 61704, (843264), ... %t A299034 Table[n! SeriesCoefficient[Product[1/(1 - x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 18}] %Y A299034 Cf. A000005, A028342, A255672, A299033. %K A299034 nonn,new %O A299034 0,3 %A A299034 _Ilya Gutkovskiy_, Feb 01 2018 %I A299033 %S A299033 1,-1,0,15,-136,885,-4896,43085,-787200,7775271,326355200, %T A299033 -22138191801,781498160640,-18924340012435,239123351330304, %U A299033 5915023788331125,-568462201562300416,25327272129182225295,-795994018378027868160,15538852668590468027711 %N A299033 a(n) = n! * [x^n] Product_{k>=1} (1 - x^k)^(n/k). %F A299033 a(n) = n! * [x^n] exp(-n*Sum_{k>=1} d(k)*x^k/k), where d(k) is the number of divisors of k (A000005). %e A299033 The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} (1 - x^k)^(n/k) begins: %e A299033 n = 0: (1), 0, 0, 0, 0, 0, 0, ... %e A299033 n = 1: 1, (-1), -1, 1, -1, 41, -131, ... %e A299033 n = 2: 1, -2, (0), 8, -4, 72, -704, ... %e A299033 n = 3: 1, -3, 3, (15), -45, 63, -1539, ... %e A299033 n = 4: 1, -4, 8, 16, (-136), 224, -1856, ... %e A299033 n = 5: 1, -5, 15, 5, -265, (885), -2075, ... %e A299033 n = 6: 1, -6, 24, -24, -396, 2376, (-4896), ... %t A299033 Table[n! SeriesCoefficient[Product[(1 - x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 19}] %Y A299033 Cf. A000005, A028343, A281267, A299034. %K A299033 sign,new %O A299033 0,4 %A A299033 _Ilya Gutkovskiy_, Feb 01 2018 %I A299032 %S A299032 1,1,0,3,6,0,12,106,420,2718,18240,120879,694320,5430438,40668264, %T A299032 300401818,2369504386,19928714475,174151735920,1543284732218, %U A299032 14224347438876,135649243229688,1331658133954940,13369350846412794,138122850643702056,1462610254141337590 %N A299032 Number of ordered ways of writing n-th triangular number as a sum of n squares of positive integers. %H A299032 Alois P. Heinz, Table of n, a(n) for n = 0..250 %H A299032 Index entries for sequences related to sums of squares %H A299032 Index to sequences related to polygonal numbers %F A299032 a(n) = [x^(n*(n+1)/2)] (Sum_{k>=1} x^(k^2))^n. %e A299032 a(4) = 6 because fourth triangular number is 10 and we have [4, 4, 1, 1], [4, 1, 4, 1], [4, 1, 1, 4], [1, 4, 4, 1], [1, 4, 1, 4] and [1, 1, 4, 4]. %p A299032 b:= proc(n, t) option remember; local i; if n=0 then %p A299032 `if`(t=0, 1, 0) elif t<1 then 0 else 0; %p A299032 for i while i^2<=n do %+b(n-i^2, t-1) od; % fi %p A299032 end: %p A299032 a:= n-> b(n*(n+1)/2, n): %p A299032 seq(a(n), n=0..25); # _Alois P. Heinz_, Feb 05 2018 %t A299032 Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n (n + 1)/2}], {n, 0, 25}] %Y A299032 Cf. A000217, A000290, A066535, A072964, A104383, A126683, A196010, A224677, A224679, A278340, A288126, A298330, A298858, A298939, A299031. %K A299032 nonn,new %O A299032 0,4 %A A299032 _Ilya Gutkovskiy_, Feb 01 2018 %I A299031 %S A299031 1,1,0,3,18,60,252,1576,10494,64152,458400,3407019,27713928,225193982, %T A299031 1980444648,17626414158,165796077562,1593587604441,15985672426992, %U A299031 163422639872978,1729188245991060,18743981599820280,208963405365941380,2378065667103672024,27742569814633730608 %N A299031 Number of ordered ways of writing n-th triangular number as a sum of n squares of nonnegative integers. %H A299031 Index entries for sequences related to sums of squares %H A299031 Index to sequences related to polygonal numbers %F A299031 a(n) = [x^(n*(n+1)/2)] (Sum_{k>=0} x^(k^2))^n. %e A299031 a(3) = 3 because third triangular number is 6 and we have [4, 1, 1], [1, 4, 1] and [1, 1, 4]. %t A299031 Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n (n + 1)/2}], {n, 0, 24}] %Y A299031 Cf. A000217, A000290, A066535, A072964, A104383, A126683, A196010, A224677, A224679, A278340, A288126, A298329, A298858, A298938, A299032. %K A299031 nonn,new %O A299031 0,4 %A A299031 _Ilya Gutkovskiy_, Feb 01 2018 %I A298989 %S A298989 1,0,1,1,2,4,8,32,101,687,3584,23564,146424,937953,6006835,38521889, %T A298989 247868209,1591813628,10234693956,65662254277,420757890998 %N A298989 Number of partitions of n^4 into fourth powers > 1. %H A298989 Eric Weisstein's World of Mathematics, Biquadratic Number %H A298989 Index entries for related partition-counting sequences %F A298989 a(n) = [x^(n^4)] Product_{k>=2} 1/(1 - x^(k^4)). %e A298989 a(4) = 2 because we have [256] and [16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16]. %Y A298989 Cf. A000583, A046042, A092362, A259793, A298641, A298859. %K A298989 nonn,more,new %O A298989 0,5 %A A298989 _Ilya Gutkovskiy_, Jan 31 2018 %I A298988 %S A298988 1,-1,0,-18,208,-2400,36504,-663754,13808320,-324176418,8487126400, %T A298988 -245122390601,7741417124880,-265402847130421,9816338228638872, %U A298988 -389618889514254225,16518399076342421248,-745025763154442071130,35619835529954597786208,-1799459812004380374518790,95780758238408017088795600 %N A298988 a(n) = [x^n] Product_{k>=1} 1/(1 + n*x^k)^k. %t A298988 Table[SeriesCoefficient[Product[1/(1 + n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 20}] %Y A298988 Cf. A255528, A261566, A261567, A266971, A292134, A297326, A298985, A298986, A298987. %K A298988 sign,new %O A298988 0,4 %A A298988 _Ilya Gutkovskiy_, Jan 31 2018 %I A298986 %S A298986 1,-1,-4,9,48,100,-756,-3479,-1600,24462,225900,364573,-643536, %T A298986 -9251736,-36989316,-32397975,165039872,1725828525,5338814652, %U A298986 8082713829,-26321848400,-233434232766,-811526778964,-1731126953532,1151302859712,23632432765000,113461901639788,287935019845749 %N A298986 a(n) = [x^n] Product_{k>=1} (1 - n*x^k)^k. %t A298986 Table[SeriesCoefficient[Product[(1 - n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 27}] %Y A298986 Cf. A073592, A266964, A292132, A297324, A298985, A298987, A298988. %K A298986 sign,new %O A298986 0,3 %A A298986 _Ilya Gutkovskiy_, Jan 31 2018 %I A299067 %S A299067 0,1,1,1,4,1,2,18,18,2,3,64,141,64,3,5,236,993,993,236,5,8,888,7330, %T A299067 13765,7330,888,8,13,3336,54106,196699,196699,54106,3336,13,21,12512, %U A299067 398654,2827609,5491159,2827609,398654,12512,21,34,46928,2937795 %N A299067 T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299067 Table starts %C A299067 ..0.....1........1..........2.............3................5..................8 %C A299067 ..1.....4.......18.........64...........236..............888...............3336 %C A299067 ..1....18......141........993..........7330............54106.............398654 %C A299067 ..2....64......993......13765........196699..........2827609...........40585250 %C A299067 ..3...236.....7330.....196699.......5491159........154373324.........4331069485 %C A299067 ..5...888....54106....2827609.....154373324.......8486867094.......465531179253 %C A299067 ..8..3336...398654...40585250....4331069485.....465531179253.....49918355153525 %C A299067 .13.12512..2937795..582407760..121483918174...25530821979245...5351692051251075 %C A299067 .21.46928.21650600.8358259950.3407860943824.1400307860660375.573807870327017333 %H A299067 R. H. Hardin, Table of n, a(n) for n = 1..199 %F A299067 Empirical for column k: %F A299067 k=1: a(n) = a(n-1) +a(n-2) %F A299067 k=2: a(n) = 4*a(n-1) -2*a(n-2) +4*a(n-3) for n>4 %F A299067 k=3: [order 9] for n>10 %F A299067 k=4: [order 22] for n>23 %F A299067 k=5: [order 62] for n>64 %e A299067 Some solutions for n=5 k=4 %e A299067 ..0..0..0..1. .0..0..1..1. .0..0..0..1. .0..0..0..0. .0..0..0..0 %e A299067 ..0..1..1..0. .0..0..1..1. .0..1..1..1. .1..0..0..1. .1..1..0..0 %e A299067 ..0..0..0..1. .1..0..0..0. .0..1..1..0. .0..1..0..1. .0..1..1..0 %e A299067 ..0..1..0..1. .1..1..1..0. .0..1..0..1. .0..1..1..0. .0..1..1..0 %e A299067 ..1..1..1..1. .0..0..1..0. .0..1..1..0. .0..0..0..0. .0..1..1..0 %Y A299067 Column 1 is A000045(n-1). %Y A299067 Column 2 is A231950(n-1). %K A299067 nonn,tabl,new %O A299067 1,5 %A A299067 _R. H. Hardin_, Feb 01 2018 %I A299066 %S A299066 8,3336,398654,40585250,4331069485,465531179253,49918355153525, %T A299066 5351692051251075,573807870327017333,61523882019695232763, %U A299066 6596584710253032969612,707285152573467933076880 %N A299066 Number of nX7 0..1 arrays with every element equal to 1, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299066 Column 7 of A299067. %H A299066 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299066 Some solutions for n=5 %e A299066 ..0..0..1..0..1..0..0. .0..0..1..0..0..0..0. .0..0..1..1..1..0..1 %e A299066 ..0..0..1..0..0..1..0. .0..0..1..0..1..1..1. .0..0..1..0..0..0..1 %e A299066 ..0..0..0..0..0..0..1. .0..0..0..0..0..0..0. .0..0..0..0..0..0..0 %e A299066 ..0..0..1..0..1..0..0. .0..0..1..0..1..1..1. .0..0..1..0..1..1..0 %e A299066 ..0..0..1..0..0..1..0. .0..0..1..0..0..1..1. .0..0..1..1..0..1..0 %Y A299066 Cf. A299067. %K A299066 nonn,new %O A299066 1,1 %A A299066 _R. H. Hardin_, Feb 01 2018 %I A299065 %S A299065 5,888,54106,2827609,154373324,8486867094,465531179253,25530821979245, %T A299065 1400307860660375,76804109131297349,4212535716237480403, %U A299065 231048259203347422103,12672487656346278900784,695058010315071776925018 %N A299065 Number of nX6 0..1 arrays with every element equal to 1, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299065 Column 6 of A299067. %H A299065 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299065 Some solutions for n=5 %e A299065 ..0..0..1..1..1..1. .0..1..0..0..0..1. .0..1..0..1..0..1. .0..1..0..0..1..1 %e A299065 ..0..0..1..1..1..1. .0..0..1..0..0..1. .0..0..1..0..0..1. .0..0..1..0..1..1 %e A299065 ..0..0..0..0..0..1. .0..0..0..1..1..1. .0..0..0..0..1..1. .0..0..0..0..1..1 %e A299065 ..0..0..1..0..1..1. .0..0..1..0..1..1. .0..0..1..1..1..1. .0..0..1..0..1..0 %e A299065 ..0..0..1..1..0..0. .0..1..0..0..0..1. .0..1..0..0..1..1. .0..0..1..0..0..1 %Y A299065 Cf. A299067. %K A299065 nonn,new %O A299065 1,1 %A A299065 _R. H. Hardin_, Feb 01 2018 %I A299064 %S A299064 3,236,7330,196699,5491159,154373324,4331069485,121483918174, %T A299064 3407860943824,95597844112736,2681715138096476,75227581448434048, %U A299064 2110287491660795753,59197879765033348597,1660621578794445613170 %N A299064 Number of nX5 0..1 arrays with every element equal to 1, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299064 Column 5 of A299067. %H A299064 R. H. Hardin, Table of n, a(n) for n = 1..210 %H A299064 R. H. Hardin, Empirical recurrence of order 62 %F A299064 Empirical recurrence of order 62 (see link above) %e A299064 Some solutions for n=5 %e A299064 ..0..0..0..0..1. .0..0..0..0..0. .0..0..0..1..0. .0..0..0..1..1 %e A299064 ..0..0..0..0..1. .0..0..0..1..0. .0..0..0..1..0. .0..0..0..0..1 %e A299064 ..1..1..1..0..1. .1..0..1..1..1. .1..1..0..0..0. .1..1..0..0..1 %e A299064 ..1..0..1..0..1. .0..1..1..0..0. .0..1..1..1..1. .0..1..1..0..1 %e A299064 ..1..1..0..1..0. .0..1..1..1..0. .0..0..0..0..1. .0..1..0..0..1 %Y A299064 Cf. A299067. %K A299064 nonn,new %O A299064 1,1 %A A299064 _R. H. Hardin_, Feb 01 2018 %I A299063 %S A299063 2,64,993,13765,196699,2827609,40585250,582407760,8358259950, %T A299063 119952006315,1721463741562,24705187860146,354550820208077, %U A299063 5088254555610537,73022914681708450,1047971560792616433 %N A299063 Number of nX4 0..1 arrays with every element equal to 1, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299063 Column 4 of A299067. %H A299063 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299063 Empirical: a(n) = 12*a(n-1) +26*a(n-2) +98*a(n-3) +216*a(n-4) -360*a(n-5) -405*a(n-6) -1419*a(n-7) -639*a(n-8) +4900*a(n-9) +2551*a(n-10) -4072*a(n-11) -2299*a(n-12) -1442*a(n-13) +1353*a(n-14) +2705*a(n-15) -1262*a(n-16) -1460*a(n-17) +621*a(n-18) +18*a(n-19) +55*a(n-20) -12*a(n-21) -4*a(n-22) for n>23 %e A299063 Some solutions for n=5 %e A299063 ..0..0..1..0. .0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..1..1 %e A299063 ..1..0..0..1. .1..0..0..1. .1..0..1..1. .1..0..0..0. .0..1..1..1 %e A299063 ..1..1..0..1. .1..1..0..1. .1..1..1..0. .0..1..1..1. .1..0..1..1 %e A299063 ..0..0..1..1. .0..0..1..0. .0..1..0..0. .1..1..1..0. .1..0..1..0 %e A299063 ..0..1..0..0. .0..0..0..1. .1..0..1..1. .1..0..0..1. .0..0..0..0 %Y A299063 Cf. A299067. %K A299063 nonn,new %O A299063 1,1 %A A299063 _R. H. Hardin_, Feb 01 2018 %I A299062 %S A299062 1,18,141,993,7330,54106,398654,2937795,21650600,159556133,1175862733, %T A299062 8665626722,63862119016,470637646700,3468406597875,25560735353400, %U A299062 188372145331363,1388225520347031,10230653220855076,75395721942360742 %N A299062 Number of nX3 0..1 arrays with every element equal to 1, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299062 Column 3 of A299067. %H A299062 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299062 Empirical: a(n) = 6*a(n-1) +8*a(n-2) +14*a(n-3) +11*a(n-4) -4*a(n-5) +3*a(n-6) +2*a(n-7) -5*a(n-8) -2*a(n-9) for n>10 %e A299062 Some solutions for n=5 %e A299062 ..0..1..0. .0..1..1. .0..1..0. .0..1..1. .0..0..0. .0..0..0. .0..1..1 %e A299062 ..1..0..1. .0..1..0. .0..0..1. .0..1..0. .0..1..0. .1..1..0. .0..0..1 %e A299062 ..0..1..1. .0..1..0. .0..1..0. .0..0..0. .0..1..1. .0..0..1. .1..1..1 %e A299062 ..1..1..0. .1..0..1. .0..1..0. .1..0..1. .1..0..1. .0..1..0. .0..1..0 %e A299062 ..1..0..1. .1..0..0. .0..0..0. .0..1..1. .0..1..1. .0..1..0. .1..0..1 %Y A299062 Cf. A299067. %K A299062 nonn,new %O A299062 1,2 %A A299062 _R. H. Hardin_, Feb 01 2018 %I A299061 %S A299061 0,4,141,13765,5491159,8486867094,49918355153525,1121607985652530679, %T A299061 96329415685984950438740,31617407509189875062181393315 %N A299061 Number of nXn 0..1 arrays with every element equal to 1, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299061 Diagonal of A299067. %e A299061 Some solutions for n=5 %e A299061 ..0..0..0..1..1. .0..0..0..0..1. .0..0..0..1..0. .0..0..0..0..0 %e A299061 ..0..0..0..1..1. .0..0..0..1..1. .0..0..0..0..1. .0..0..0..1..0 %e A299061 ..1..0..1..0..0. .1..0..1..0..1. .0..1..1..1..0. .1..0..1..0..1 %e A299061 ..0..1..0..0..1. .0..1..0..0..1. .0..1..1..0..1. .0..1..1..0..1 %e A299061 ..0..0..1..0..1. .1..0..0..0..0. .0..0..1..0..0. .0..1..1..1..1 %Y A299061 Cf. A299067. %K A299061 nonn,new %O A299061 1,2 %A A299061 _R. H. Hardin_, Feb 01 2018 %I A299060 %S A299060 1,1,1,1,5,1,1,13,13,1,1,42,38,42,1,1,127,199,199,127,1,1,389,864, %T A299060 2096,864,389,1,1,1192,4366,17930,17930,4366,1192,1,1,3645,21804, %U A299060 175031,295407,175031,21804,3645,1,1,11161,111861,1718789,5558665,5558665 %N A299060 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299060 Table starts %C A299060 .1....1......1........1..........1............1..............1................1 %C A299060 .1....5.....13.......42........127..........389...........1192.............3645 %C A299060 .1...13.....38......199........864.........4366..........21804...........111861 %C A299060 .1...42....199.....2096......17930.......175031........1718789.........17101477 %C A299060 .1..127....864....17930.....295407......5558665......104819165.......1992248544 %C A299060 .1..389...4366...175031....5558665....199153610.....7165136807.....259511787831 %C A299060 .1.1192..21804..1718789..104819165...7165136807...491688804213...33934925502793 %C A299060 .1.3645.111861.17101477.1992248544.259511787831.33934925502793.4462148033879670 %H A299060 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299060 Empirical for column k: %F A299060 k=1: a(n) = a(n-1) %F A299060 k=2: a(n) = a(n-1) +5*a(n-2) +4*a(n-3) %F A299060 k=3: [order 14] for n>16 %F A299060 k=4: [order 38] for n>39 %e A299060 Some solutions for n=5 k=4 %e A299060 ..0..0..1..0. .0..0..1..1. .0..0..1..1. .0..0..1..0. .0..1..0..0 %e A299060 ..0..1..1..1. .0..0..1..1. .1..0..1..1. .0..0..1..1. .0..0..0..1 %e A299060 ..1..1..0..1. .1..0..0..0. .0..0..1..0. .0..0..1..1. .1..1..0..0 %e A299060 ..1..1..1..1. .1..1..0..1. .1..0..0..0. .0..0..0..1. .0..1..1..0 %e A299060 ..0..1..0..1. .1..1..1..1. .0..0..0..1. .0..1..0..0. .0..0..0..0 %Y A299060 Column 2 is A298234. %K A299060 nonn,tabl,new %O A299060 1,5 %A A299060 _R. H. Hardin_, Feb 01 2018 %I A299059 %S A299059 1,1192,21804,1718789,104819165,7165136807,491688804213, %T A299059 33934925502793,2354116956503187,163395008821942912, %U A299059 11352308319678540454,788903049713608829357,54831472652793505844543,3811221344535417966318764 %N A299059 Number of nX7 0..1 arrays with every element equal to 0, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299059 Column 7 of A299060. %H A299059 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299059 Some solutions for n=5 %e A299059 ..0..0..1..1..1..0..0. .0..0..1..1..0..0..0. .0..0..1..0..1..0..1 %e A299059 ..0..0..1..0..1..1..0. .0..0..1..0..0..1..0. .0..0..1..1..1..0..0 %e A299059 ..0..0..0..1..0..0..0. .0..0..0..1..1..1..1. .0..0..0..1..1..0..1 %e A299059 ..0..0..1..0..1..0..0. .0..0..1..0..0..1..0. .0..0..1..0..1..1..1 %e A299059 ..1..0..1..1..0..0..1. .1..0..1..1..0..1..1. .1..0..0..1..1..1..0 %Y A299059 Cf. A299060. %K A299059 nonn,new %O A299059 1,2 %A A299059 _R. H. Hardin_, Feb 01 2018 %I A299058 %S A299058 1,389,4366,175031,5558665,199153610,7165136807,259511787831, %T A299058 9448796539484,344252167371786,12555436461782784,458025124087582401, %U A299058 16711604849240776092,609784948993079811760,22250906791148467546626 %N A299058 Number of nX6 0..1 arrays with every element equal to 0, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299058 Column 6 of A299060. %H A299058 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299058 Some solutions for n=5 %e A299058 ..0..0..1..1..1..0. .0..0..0..0..0..0. .0..0..1..0..0..0. .0..0..1..1..0..0 %e A299058 ..0..0..1..1..1..1. .0..0..1..1..0..1. .0..0..1..1..0..1. .0..0..1..0..0..0 %e A299058 ..0..0..1..0..0..1. .0..0..1..0..0..0. .0..0..0..0..1..1. .0..0..1..0..1..1 %e A299058 ..0..1..0..1..0..0. .0..1..0..0..1..1. .0..1..0..1..0..1. .0..1..0..0..1..1 %e A299058 ..1..1..1..1..1..0. .1..1..0..0..1..0. .0..0..0..0..1..1. .0..0..1..1..1..1 %Y A299058 Cf. A299060. %K A299058 nonn,new %O A299058 1,2 %A A299058 _R. H. Hardin_, Feb 01 2018 %I A299057 %S A299057 1,127,864,17930,295407,5558665,104819165,1992248544,38074261228, %T A299057 728095163211,13939203627469,266922192325475,5112200956467065, %U A299057 97917633017720051,1875541645351871211,35925207100912996734 %N A299057 Number of nX5 0..1 arrays with every element equal to 0, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299057 Column 5 of A299060. %H A299057 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299057 Some solutions for n=5 %e A299057 ..0..0..0..0..0. .0..0..0..1..1. .0..0..0..1..1. .0..0..0..1..0 %e A299057 ..1..0..0..0..1. .1..0..0..1..1. .1..0..1..1..0. .1..0..0..0..0 %e A299057 ..1..1..1..1..1. .1..1..0..1..0. .1..1..1..0..0. .0..0..1..1..0 %e A299057 ..0..1..0..1..0. .0..0..0..1..1. .1..0..0..0..0. .1..1..1..1..1 %e A299057 ..0..0..0..0..0. .0..0..1..1..0. .0..0..1..0..0. .0..1..1..0..1 %Y A299057 Cf. A299060. %K A299057 nonn,new %O A299057 1,2 %A A299057 _R. H. Hardin_, Feb 01 2018 %I A299056 %S A299056 1,42,199,2096,17930,175031,1718789,17101477,171333994,1720010217, %T A299056 17294372389,173985722475,1750874896463,17622070573506, %U A299056 177372546408622,1785376032768684,17971288389656990,180897169816432156 %N A299056 Number of nX4 0..1 arrays with every element equal to 0, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299056 Column 4 of A299060. %H A299056 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299056 Empirical: a(n) = 8*a(n-1) +46*a(n-2) -154*a(n-3) -1109*a(n-4) +120*a(n-5) +8740*a(n-6) +9747*a(n-7) -21243*a(n-8) -32715*a(n-9) +35745*a(n-10) +30176*a(n-11) -79460*a(n-12) -60396*a(n-13) +168083*a(n-14) +59241*a(n-15) -212900*a(n-16) -111057*a(n-17) +602692*a(n-18) -96590*a(n-19) -518453*a(n-20) +106519*a(n-21) +395496*a(n-22) +82173*a(n-23) -470675*a(n-24) -282212*a(n-25) +137909*a(n-26) +870028*a(n-27) -989190*a(n-28) +390933*a(n-29) -423269*a(n-30) +512444*a(n-31) +29537*a(n-32) -223600*a(n-33) +8415*a(n-34) -55396*a(n-35) +59174*a(n-36) +11618*a(n-37) -9924*a(n-38) for n>39 %e A299056 Some solutions for n=5 %e A299056 ..0..0..0..0. .0..0..1..1. .0..1..1..0. .0..1..1..0. .0..0..1..1 %e A299056 ..1..0..1..0. .0..0..1..1. .0..0..1..1. .1..1..0..0. .0..0..1..1 %e A299056 ..0..0..0..0. .0..0..0..0. .0..0..1..0. .0..1..0..0. .1..1..1..1 %e A299056 ..1..0..1..0. .1..1..1..0. .1..1..0..0. .0..0..1..1. .1..0..1..1 %e A299056 ..0..0..1..1. .1..1..1..1. .1..1..0..0. .0..0..0..1. .0..0..1..1 %Y A299056 Cf. A299060. %K A299056 nonn,new %O A299056 1,2 %A A299056 _R. H. Hardin_, Feb 01 2018 %I A299055 %S A299055 1,13,38,199,864,4366,21804,111861,578509,3007390,15688017,81963177, %T A299055 428662899,2243048643,11740648591,61463549876,321797679318, %U A299055 1684885835098,8822066509377,46193090987303,241873068747484,1266485180192256 %N A299055 Number of nX3 0..1 arrays with every element equal to 0, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299055 Column 3 of A299060. %H A299055 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299055 Empirical: a(n) = 5*a(n-1) +9*a(n-2) -31*a(n-3) -62*a(n-4) +46*a(n-5) +78*a(n-6) -36*a(n-7) +53*a(n-8) +101*a(n-9) -86*a(n-10) -14*a(n-11) -26*a(n-12) -28*a(n-13) +6*a(n-14) for n>16 %e A299055 Some solutions for n=5 %e A299055 ..0..1..1. .0..1..0. .0..0..1. .0..1..0. .0..0..1. .0..1..0. .0..0..1 %e A299055 ..1..1..0. .0..0..0. .0..1..1. .1..1..1. .0..1..1. .1..1..1. .0..1..1 %e A299055 ..1..0..0. .1..0..0. .1..1..0. .0..1..0. .0..0..1. .1..1..0. .0..0..1 %e A299055 ..0..0..1. .0..0..1. .1..1..1. .0..0..0. .1..0..0. .0..0..0. .1..0..0 %e A299055 ..0..0..0. .0..1..1. .0..1..0. .1..0..1. .0..0..1. .1..0..0. .0..0..0 %Y A299055 Cf. A299060. %K A299055 nonn,new %O A299055 1,2 %A A299055 _R. H. Hardin_, Feb 01 2018 %I A299054 %S A299054 1,5,38,2096,295407,199153610,491688804213,4462148033879670, %T A299054 148475436881582614613 %N A299054 Number of nXn 0..1 arrays with every element equal to 0, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299054 Diagonal of A299060. %e A299054 Some solutions for n=5 %e A299054 ..0..0..0..0..0. .0..0..0..1..0. .0..0..0..1..1. .0..0..0..1..1 %e A299054 ..1..0..1..1..0. .0..0..0..1..1. .0..1..1..0..1. .0..1..1..0..1 %e A299054 ..0..0..1..0..0. .1..0..1..1..0. .1..1..0..0..0. .1..0..0..0..1 %e A299054 ..0..0..0..1..1. .0..0..0..1..1. .1..1..1..1..1. .1..0..1..1..1 %e A299054 ..0..0..1..1..0. .0..0..0..1..0. .0..1..1..0..1. .1..1..1..1..1 %Y A299054 Cf. A299060. %K A299054 nonn,new %O A299054 1,2 %A A299054 _R. H. Hardin_, Feb 01 2018 %I A299052 %S A299052 1,2,2,3,4,3,5,4,4,5,8,16,11,16,8,13,50,40,40,50,13,21,112,79,455,79, %T A299052 112,21,34,348,480,1650,1650,480,348,34,55,1028,1542,10706,9994,10706, %U A299052 1542,1028,55,89,2796,5317,76068,82340,82340,76068,5317,2796,89,144,8216 %N A299052 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299052 Table starts %C A299052 ..1....2.....3.......5........8.........13...........21.............34 %C A299052 ..2....4.....4......16.......50........112..........348...........1028 %C A299052 ..3....4....11......40.......79........480.........1542...........5317 %C A299052 ..5...16....40.....455.....1650......10706........76068.........445597 %C A299052 ..8...50....79....1650.....9994......82340.......953445........8956381 %C A299052 .13..112...480...10706....82340....1731008.....29176924......456270494 %C A299052 .21..348..1542...76068...953445...29176924....888675946....23405238642 %C A299052 .34.1028..5317..445597..8956381..456270494..23405238642..1063108306477 %C A299052 .55.2796.22571.2902533.86708195.8295241404.693920505214.53324500523781 %H A299052 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299052 Empirical for column k: %F A299052 k=1: a(n) = a(n-1) +a(n-2) %F A299052 k=2: a(n) = 2*a(n-1) +2*a(n-2) +6*a(n-3) -10*a(n-4) -8*a(n-5) for n>6 %F A299052 k=3: [order 17] for n>18 %F A299052 k=4: [order 55] for n>58 %e A299052 Some solutions for n=5 k=4 %e A299052 ..0..0..1..0. .0..0..1..0. .0..1..0..0. .0..1..0..0. .0..0..1..0 %e A299052 ..0..0..1..0. .0..0..0..1. .1..0..0..0. .1..0..0..0. .0..0..0..1 %e A299052 ..0..0..1..1. .0..1..1..1. .1..1..0..0. .1..1..0..0. .1..1..0..0 %e A299052 ..0..1..1..0. .0..0..0..1. .1..1..1..0. .1..1..0..1. .1..1..0..0 %e A299052 ..0..1..1..0. .1..0..0..1. .0..1..0..1. .0..1..1..0. .0..1..1..1 %Y A299052 Column 1 is A000045(n+1). %Y A299052 Column 2 is A298148. %K A299052 nonn,tabl,new %O A299052 1,2 %A A299052 _R. H. Hardin_, Feb 01 2018 %I A299051 %S A299051 21,348,1542,76068,953445,29176924,888675946,23405238642,693920505214, %T A299051 21037340897708,603764830153053,17934063108751557,534096869604892352, %U A299051 15674627896951182550,464308514446686398318,13757814178967296681530 %N A299051 Number of nX7 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299051 Column 7 of A299052. %H A299051 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299051 Some solutions for n=5 %e A299051 ..0..0..1..1..1..0..1. .0..1..1..1..1..0..1. .0..1..1..0..1..0..1 %e A299051 ..0..0..1..1..1..1..0. .0..1..1..0..1..1..0. .0..1..1..0..1..0..1 %e A299051 ..0..0..0..0..1..1..1. .0..0..0..0..0..0..0. .0..0..0..0..0..0..0 %e A299051 ..0..0..1..0..0..0..1. .0..1..1..1..1..1..0. .0..1..1..1..0..0..0 %e A299051 ..1..0..0..1..0..0..1. .0..1..1..1..1..1..0. .1..0..1..0..1..0..1 %Y A299051 Cf. A299052. %K A299051 nonn,new %O A299051 1,1 %A A299051 _R. H. Hardin_, Feb 01 2018 %I A299050 %S A299050 13,112,480,10706,82340,1731008,29176924,456270494,8295241404, %T A299050 145705046354,2478160069319,43916073524508,769004834004139, %U A299050 13362142854536483,234592273093074307,4104754183842510101 %N A299050 Number of nX6 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299050 Column 6 of A299052. %H A299050 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299050 Some solutions for n=5 %e A299050 ..0..0..1..0..0..1. .0..1..1..0..1..0. .0..1..1..0..1..1. .0..1..0..0..0..1 %e A299050 ..0..0..0..1..0..0. .0..1..1..1..1..1. .1..1..0..1..1..1. .1..0..0..1..0..0 %e A299050 ..0..1..1..1..1..0. .1..1..1..0..1..0. .1..1..0..0..0..0. .0..0..0..0..0..0 %e A299050 ..0..0..0..1..0..0. .1..0..1..0..1..1. .1..1..0..0..1..1. .1..0..1..1..0..0 %e A299050 ..0..0..1..0..0..0. .0..1..1..1..1..1. .0..1..1..1..1..1. .0..1..1..1..0..0 %Y A299050 Cf. A299052. %K A299050 nonn,new %O A299050 1,1 %A A299050 _R. H. Hardin_, Feb 01 2018 %I A299049 %S A299049 8,50,79,1650,9994,82340,953445,8956381,86708195,930107134,9283822089, %T A299049 93868491793,972801539169,9876709646807,100715230941566, %U A299049 1033318335538221,10540518267901855,107677643657416903,1101639196235269011 %N A299049 Number of nX5 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299049 Column 5 of A299052. %H A299049 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299049 Some solutions for n=5 %e A299049 ..0..1..0..0..1. .0..1..1..1..1. .0..0..0..0..1. .0..1..1..1..1 %e A299049 ..0..1..0..0..0. .1..1..1..0..0. .0..0..0..1..0. .1..1..0..1..1 %e A299049 ..1..1..1..0..0. .1..1..0..0..0. .1..1..1..1..1. .1..1..1..1..0 %e A299049 ..1..1..1..1..1. .1..0..0..1..1. .1..1..0..1..1. .1..1..0..1..1 %e A299049 ..0..0..1..0..0. .1..0..0..1..1. .0..1..1..1..0. .1..1..0..1..1 %Y A299049 Cf. A299052. %K A299049 nonn,new %O A299049 1,1 %A A299049 _R. H. Hardin_, Feb 01 2018 %I A299048 %S A299048 5,16,40,455,1650,10706,76068,445597,2902533,19204880,121652513, %T A299048 790850405,5141909319,33135126906,214855459081,1392224798269, %U A299048 9005556603052,58337331135135,377789663876767,2445748153198451 %N A299048 Number of nX4 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299048 Column 4 of A299052. %H A299048 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299048 Empirical: a(n) = 4*a(n-1) +17*a(n-2) +91*a(n-3) -441*a(n-4) -1412*a(n-5) -1795*a(n-6) +14444*a(n-7) +38743*a(n-8) +4354*a(n-9) -219442*a(n-10) -445582*a(n-11) +215939*a(n-12) +1755120*a(n-13) +1856749*a(n-14) -2851789*a(n-15) -7056619*a(n-16) +1896568*a(n-17) +16599843*a(n-18) +7411988*a(n-19) -36570210*a(n-20) -48189209*a(n-21) +30887758*a(n-22) +117508143*a(n-23) +66164602*a(n-24) -107676097*a(n-25) -194033556*a(n-26) -32507518*a(n-27) +189427489*a(n-28) +153275440*a(n-29) -48740284*a(n-30) -186966176*a(n-31) -55538244*a(n-32) +153463562*a(n-33) +178445765*a(n-34) -74905690*a(n-35) -229954672*a(n-36) -76342848*a(n-37) +126582799*a(n-38) +88979427*a(n-39) -6412703*a(n-40) -3164622*a(n-41) -2717580*a(n-42) -20660957*a(n-43) -20947988*a(n-44) +5951972*a(n-45) +11311952*a(n-46) +1501468*a(n-47) -813181*a(n-48) -801514*a(n-49) -581210*a(n-50) -130320*a(n-51) +42224*a(n-52) +57304*a(n-53) +22016*a(n-54) +2560*a(n-55) for n>58 %e A299048 Some solutions for n=5 %e A299048 ..0..0..1..0. .0..0..0..0. .0..0..0..0. .0..1..0..0. .0..0..0..1 %e A299048 ..1..1..1..1. .0..0..0..0. .0..0..1..1. .1..0..0..0. .1..1..0..0 %e A299048 ..0..0..1..1. .1..0..1..1. .0..0..1..1. .1..1..0..0. .1..1..0..0 %e A299048 ..0..0..1..1. .0..0..1..1. .0..0..1..0. .1..1..1..0. .1..1..1..0 %e A299048 ..1..1..1..1. .1..0..0..0. .0..0..1..0. .0..1..0..1. .0..1..1..0 %Y A299048 Cf. A299052. %K A299048 nonn,new %O A299048 1,1 %A A299048 _R. H. Hardin_, Feb 01 2018 %I A299047 %S A299047 3,4,11,40,79,480,1542,5317,22571,80346,297210,1158279,4266972, %T A299047 16017750,60820802,227040843,853970252,3218688708,12071422085, %U A299047 45398669314,170767134707,641450486645,2411776993812,9067212243589,34076147843251 %N A299047 Number of nX3 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299047 Column 3 of A299052. %H A299047 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299047 Empirical: a(n) = 2*a(n-1) +5*a(n-2) +21*a(n-3) -39*a(n-4) -61*a(n-5) -56*a(n-6) +110*a(n-7) +183*a(n-8) +50*a(n-9) -27*a(n-10) -125*a(n-11) -64*a(n-12) -141*a(n-13) +36*a(n-14) +5*a(n-15) +8*a(n-16) +6*a(n-17) for n>18 %e A299047 Some solutions for n=5 %e A299047 ..0..1..0. .0..0..0. .0..1..1. .0..0..0. .0..1..1. .0..1..1. .0..0..1 %e A299047 ..1..1..1. .0..0..0. .0..1..1. .0..0..0. .0..1..1. .0..1..1. .0..0..0 %e A299047 ..1..1..1. .1..1..0. .0..0..0. .1..0..1. .0..0..0. .1..1..1. .0..1..0 %e A299047 ..0..0..1. .0..0..0. .0..1..1. .0..0..0. .0..0..0. .1..1..0. .0..0..0 %e A299047 ..0..0..1. .0..0..0. .0..1..1. .1..0..0. .1..0..1. .1..1..0. .0..0..1 %Y A299047 Cf. A299052. %K A299047 nonn,new %O A299047 1,1 %A A299047 _R. H. Hardin_, Feb 01 2018 %I A299046 %S A299046 1,4,11,455,9994,1731008,888675946,1063108306477,4680605638885458 %N A299046 Number of nXn 0..1 arrays with every element equal to 0, 1, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299046 Diagonal of A299052. %e A299046 Some solutions for n=5 %e A299046 ..0..1..0..1..0. .0..1..1..1..0. .0..0..0..0..1. .0..1..0..0..1 %e A299046 ..0..1..0..1..0. .1..0..1..1..1. .0..0..1..0..0. .1..0..0..1..0 %e A299046 ..0..0..0..1..1. .0..0..0..1..0. .1..0..1..0..1. .1..1..0..0..0 %e A299046 ..1..0..0..1..0. .1..0..1..1..1. .0..0..0..0..0. .1..1..1..0..0 %e A299046 ..0..1..0..0..1. .1..0..1..1..0. .0..0..1..0..0. .0..1..1..1..1 %Y A299046 Cf. A299052. %K A299046 nonn,new %O A299046 1,2 %A A299046 _R. H. Hardin_, Feb 01 2018 %I A295766 %S A295766 1,1,5,90,3204,170987,12162683,1087504130,118227836360,15304211345298, %T A295766 2324856843115770,409872125913866852,83092182794794380856, %U A295766 19214014336799266619671,5030971580159960051721815,1481724835890098667273954338,487883202104697456579537247232,178595806151469762148235569612814,72312528698655521190143801630975174 %N A295766 G.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2)/n^2 = [x^(n-2)] A(x)^(n^2) for n>=2 with A'(0) = 1. %C A295766 Compare g.f. to: [x^(n-1)] G(x)^(n^2)/n^2 = [x^(n-2)] G(x)^(n^2)/(n-1) for n>=2 holds when G(x) = exp(x). %H A295766 Paul D. Hanna, Table of n, a(n) for n = 0..260 %F A295766 a(A075427(k) - 1) is odd for n>=0 and a(n) is even elsewhere (conjecture). %e A295766 G.f.: A(x) = 1 + x + 5*x^2 + 90*x^3 + 3204*x^4 + 170987*x^5 + 12162683*x^6 + 1087504130*x^7 + 118227836360*x^8 + 15304211345298*x^9 + 2324856843115770*x^10 + ... %e A295766 ILLUSTRATION OF THE DEFINITION. %e A295766 The table of coefficients of x^k in A(x)^(n^2) begins: %e A295766 n=1: [1, 1, 5, 90, 3204, 170987, 12162683, ...]; %e A295766 n=2: [1, 4, 26, 424, 14107, 729196, 50993674, ...]; %e A295766 n=3: [1, 9, 81, 1254, 37602, 1833597, 124332453, ...]; %e A295766 n=4: [1, 16, 200, 3200, 86084, 3846720, 248466736, ...]; %e A295766 n=5: [1, 25, 425, 7550, 188750, 7566705, 455263225, ...]; %e A295766 n=6: [1, 36, 810, 16680, 410499, 14777964, 808802730, ...]; %e A295766 n=7: [1, 49, 1421, 34594, 886312, 29473255, 1444189495, ...]; ... %e A295766 in which the main diagonal %e A295766 [1, 4, 81, 3200, 188750, 14777964, 1444189495, ...] %e A295766 is related to an adjacent diagonal by dividing by n^2 like so: %e A295766 [1, 4/4, 81/9, 3200/16, 188750/25, 14777964/36, 1444189495/49, ...] %e A295766 = [1, 1, 9, 200, 7550, 410499, 29473255, ...]. %e A295766 Thus [x^(n-1)] A(x)^(n^2)/n^2 = [x^(n-2)] A(x)^(n^2) for n>=2. %o A295766 (PARI) {a(n) = my(A=[1],V); for(m=2,n+1, A=concat(A,0); V=Vec(Ser(A)^(m^2)); A[#A] = V[#A-1] - V[#A]/m^2 );A[n+1]} %o A295766 for(n=0,20,print1(a(n),", ")) %o A295766 (PARI) /* Informal method of obtaining N terms: */ %o A295766 N=30; A=[1]; for(n=2,N, A=concat(A,0); V=Vec(Ser(A)^(n^2)); A[#A] = V[#A-1] - V[#A]/n^2 );A %Y A295766 Cf. A088715, A295811, A075427. %K A295766 nonn,new %O A295766 0,3 %A A295766 _Paul D. Hanna_, Jan 31 2018 %I A299018 %S A299018 1,2,2,6,11,6,24,60,60,24,120,366,501,366,120,720,2532,4242,4242,2532, %T A299018 720,5040,19764,38268,46863,38268,19764,5040,40320,172512,373104, %U A299018 528336,528336,373104,172512,40320,362880,1668528,3942108,6237828,7213761,6237828,3942108,1668528,362880 %N A299018 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial P(n) = n*(x + 1)*P(n - 1) - (n - 2)^2*x*P(n - 2). %F A299018 P(0) = 0, P(1) = 1 and P(n) = n * (x + 1) * P(n - 1) - (n - 2)^2 * x * P(n - 2). %e A299018 For n = 3, the polynomial is 6*x^2 + 11*x + 6. %e A299018 The first few polynomials, as a table: %e A299018 [1], %e A299018 [2, 2], %e A299018 [6, 11, 6], %e A299018 [24, 60, 60, 24], %e A299018 [120, 366, 501, 366, 120] %p A299018 P:= proc(n) option remember; expand(`if`(n<2, n, %p A299018 n*(x+1)*P(n-1)-(n-2)^2*x*P(n-2))) %p A299018 end: %p A299018 T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(P(n)): %p A299018 seq(T(n), n=1..12); # _Alois P. Heinz_, Jan 31 2018 %o A299018 (Sage) %o A299018 @cached_function %o A299018 def poly(n): %o A299018 x = polygen(ZZ, 'x') %o A299018 if n < 1: %o A299018 return x.parent().zero() %o A299018 elif n == 1: %o A299018 return x.parent().one() %o A299018 else: %o A299018 return n * (x + 1) * poly(n - 1) - (n - 2)**2 * x * poly(n - 2) %Y A299018 Very similar to A298854. %Y A299018 Row sums are A277382(n-1) for n>0. %Y A299018 Leftmost and rightmost columns are A000142. %Y A299018 Alternating row sums are A177145. %Y A299018 Alternating row sum of row 2*n+1 is A001818(n). %K A299018 tabl,nonn,easy,new %O A299018 1,2 %A A299018 _F. Chapoton_, Jan 31 2018 %I A297306 %S A297306 7,43,79,163,673,853,919,1063,1429,1549,1663,2143,2683,3229,3499,4993, %T A297306 5119,5653,5779,6229,6343,7333,7459,7669,8353,8539,8719,9829,10009, %U A297306 10243,10303,11383,11689,12583,13399,14149,14653,14923,15649,16603,17053,17389,17749 %N A297306 Primes p such that q = 4*p+1 and r = (2*p+1)/3 are also primes. %C A297306 This sequence was suggested by _Moshe Shmuel Newman_. It has its source in his study of finite groups. %H A297306 Alois P. Heinz, Table of n, a(n) for n = 1..10000 %e A297306 Prime p = 7 is in the sequence because q = 4*7+1 = 29 and r = (2*7+1)/3 = 5 are also primes. %p A297306 a:= proc(n) option remember; local p; p:= `if`(n=1, 1, a(n-1)); %p A297306 do p:= nextprime(p); if irem(p, 3)=1 and %p A297306 isprime(4*p+1) and isprime((2*p+1)/3) then break fi %p A297306 od; p %p A297306 end: %p A297306 seq(a(n), n=1..50); # _Alois P. Heinz_, Jan 07 2018 %Y A297306 Cf. A000040. %K A297306 nonn,new %O A297306 1,1 %A A297306 _David S. Newman_, Jan 04 2018 %E A297306 More terms from _Alois P. Heinz_, Jan 07 2018 %I A298799 %S A298799 2,4,28,108,2,13968,480,7914054,433284,18726123500,256, %T A298799 178290006448984,14454384 %N A298799 Number of optimal solutions to the maximal number of diagonals problem studied in A264041. %Y A298799 Cf. A264041. %K A298799 nonn,more,new %O A298799 1,1 %A A298799 _Rob Pratt_, Nov 09 2015 %E A298799 a(8) corrected and a(9)-a(13) from _Andrew Howroyd_, Feb 03 2018 %I A298951 %S A298951 13,673,1595813,492366587,9809862296159 %N A298951 Wieferich primes to base 22. %C A298951 Prime numbers p such that p^2 divides 22^(p-1) - 1. %C A298951 Next term, if it exists, is larger than 8.72*10^13. %C A298951 492366587 was found by Montgomery (cf. Montgomery, 1993). - _Felix Fröhlich_, Jan 30 2018 %H A298951 Richard Fischer, Fermatquotient B^(P-1) == 1 (mod P^2) %H A298951 P. L. Montgomery, New Solutions of a^p-1 == 1 (mod p^2), Mathematics of Computation, Vol. 61, No. 203 (1993), 361-363. %H A298951 Wikipedia, Wieferich prime %o A298951 (PARI) forprime(p=1, , if(Mod(22, p^2)^(p-1)==1, print1(p, ", "))) %Y A298951 Cf. A001220, A014127, A123692, A212583, A123693, A045616, A111027, A128667, A234810, A242741, A128668, A244260, A090968, A242982, A128669. %K A298951 nonn,more,hard,new %O A298951 1,1 %A A298951 _Tim Johannes Ohrtmann_, Jan 30 2018 %I A299016 %S A299016 7,9,13,19,37,63,79,97,117,139,163,217,247,279,313,349,387,427,469, %T A299016 559,607,657,709,763,819,877,937,1063,1129,1197,1267,1339,1413,1489, %U A299016 1567,1729,1899,1987,2077,2169,2263,2359,2557,2659,2763,2869,2977,3199,3313,3547,3667,3789,3913,4039,4167,4297,4429,4699,4837,4977 %N A299016 Numbers n such that the cubic polynomial x^3 - n*x - n has a square discriminant. %e A299016 7 gives the algebraic space of heptagons, 9 gives the algebraic space of nonagons. %t A299016 Last/@Select[Table[{NumberFieldDiscriminant[Root[-n-n #1+#1^3&,1]],n^2, n},{n,1,5000}],#[[1]]==#[[2]]&] %o A299016 (PARI) isok(n) = issquare(poldisc(x^3-n*x-n)); \\ _Michel Marcus_, Jan 31 2018 %K A299016 nonn,easy,new %O A299016 1,1 %A A299016 _Ed Pegg Jr_, Jan 31 2018 %E A299016 Name edited by _Michel Marcus_, Feb 02 2018 %I A296106 %S A296106 1,3,3,8,17,8,21,130,130,21,55,931,2604,931,55,144,6871,54732,54732, %T A296106 6871,144,377,50778 %N A296106 Square array T(n,k) n >= 1, k >= 1 read by antidiagonals: T(n, k) is the number of distinct Bojagi boards with dimensions n X k that have a unique solution. %C A296106 Bojagi is a puzzle game created by David Radcliffe. %C A296106 A Bojagi board is a rectangular board with some cells empty and some cells containing positive integers. A solution for a Bojagi board partitions the board into rectangles such that each rectangle contains exactly one integer, and that integer is the area of the rectangle. %H A296106 Taotao Liu, Thomas Ledbetter C# Program %H A296106 David Radcliffe, Rules of puzzle game Bojagi %F A296106 T(n,1) = A088305(n), the even-indexed Fibonacci numbers. %F A296106 T(n,1) = Sum_{i=1..n} i*T(n-i,1) if we take T(0,1) = 1. %e A296106 Array begins: %e A296106 ====================================== %e A296106 n\k| 1 2 3 4 5 6 %e A296106 ---+---------------------------------- %e A296106 1 | 1 3 8 21 55 144 ... %e A296106 2 | 3 17 130 931 6871 ... %e A296106 3 | 8 130 2604 54732 ... %e A296106 4 | 21 931 54732 ... %e A296106 5 | 55 6871 ... %e A296106 6 | 144 ... %e A296106 ... %e A296106 As a triangle: %e A296106 1; %e A296106 3, 3; %e A296106 8, 17, 8; %e A296106 21, 130, 130, 21; %e A296106 55, 931, 2604, 931, 55; %e A296106 144, 6871, 54732, 54732, 6871, 144; %e A296106 ... %e A296106 If n=1 or k=1, any valid board (a board whose numbers add up to the area of the board) has a unique solution. %e A296106 For n=2 and k=2, there are 17 boards that have a unique solution. There is 1 board in which each of the four cells has a 1. %e A296106 There are 4 boards which contain two 2's. The 2's must be adjacent (not diagonally opposite) in order for the board to have a unique solution. %e A296106 There are 8 boards which contain one 2 and two 1's. The 1's must be adjacent in order for the board to have a solution. The 2 can be placed in either of the remaining two cells. %e A296106 There are 4 boards which contain one 4. It can be placed anywhere. %Y A296106 Cf. A088305. %K A296106 hard,nonn,tabl,more,new %O A296106 1,2 %A A296106 _Taotao Liu_, Dec 04 2017 %I A298985 %S A298985 1,1,8,54,496,5400,73728,1204322,23167808,512093178,12781430600, %T A298985 355128859129,10863077554224,362572265689777,13107541496092960, %U A298985 510105773344747725,21258690342206888192,944467894258279964254,44555341678790400325512,2224158766859058600584834,117123916650423288611260400 %N A298985 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k)^k. %H A298985 Vaclav Kotesovec, Table of n, a(n) for n = 0..380 %F A298985 a(n) ~ n^n. - _Vaclav Kotesovec_, Feb 02 2018 %t A298985 Table[SeriesCoefficient[Product[1/(1 - n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 20}] %Y A298985 Cf. A000219, A124577, A261561, A261565, A266941, A297329, A298986, A298987, A298988. %K A298985 nonn,new %O A298985 0,3 %A A298985 _Ilya Gutkovskiy_, Jan 31 2018 %I A298987 %S A298987 1,1,4,27,80,400,1908,6223,31296,116478,450100,1828915,7360848, %T A298987 26906828,95776772,403908975,1421758720,5072014447,18481180644, %U A298987 68350964211,246180936400,827642046294,2958748580084,10294629775620,36607347335232,120800714172500,407951731319860,1405943613730899 %N A298987 a(n) = [x^n] Product_{k>=1} (1 + n*x^k)^k. %H A298987 Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 %t A298987 Table[SeriesCoefficient[Product[(1 + n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 27}] %Y A298987 Cf. A026007, A261562, A266857, A266891, A291698, A297322, A298985, A298986, A298988. %K A298987 nonn,new %O A298987 0,3 %A A298987 _Ilya Gutkovskiy_, Jan 31 2018 %I A298994 %S A298994 1,2,6,52,134,956,4124,20008,73158,439660,1874612,8350808,37583004, %T A298994 169862616,779948152,3774085968,15435601222,69542934604,329825707332, %U A298994 1403190752632,6313190864052,29079505547912,126937389732872,552273916408368,2477249228318748 %N A298994 Expansion of Product_{n>=1} (1 + (4*x)^n)^(1/2). %H A298994 Seiichi Manyama, Table of n, a(n) for n = 0..1000 %F A298994 Convolution inverse of A298993. %Y A298994 Cf. A298411, A298993. %K A298994 nonn,new %O A298994 0,2 %A A298994 _Seiichi Manyama_, Jan 31 2018 %I A298993 %S A298993 1,-2,-2,-36,54,-476,556,-6088,35878,-156844,444164,-1734648,11948604, %T A298993 -35313048,156354328,-864527760,4733447686,-12692853452,54065039380, %U A298993 -226098757912,1278838329812,-5257771138376,19455009120232,-76455773381360,453306681446748 %N A298993 Expansion of Product_{n>=1} 1/sqrt(1 + (4*x)^n). %Y A298993 Cf. A000984, A271235, A298994. %K A298993 sign,new %O A298993 0,2 %A A298993 _Seiichi Manyama_, Jan 31 2018 %I A298990 %S A298990 1,2,4,5,8,11,12,17,18,37,24,53,30,89,39,71,42,101,45,179,57,137,72, %T A298990 193,60,233,84,257,90,251,117,401,123,311,144,373,120,347,105,457,162, %U A298990 661,150,479,180,547,237,599,165,617,264,641,288,683,195,907,231,881 %N A298990 Least number m such that A035026(m) = n. %C A298990 The upper branch visible in the scatterplot of the first terms corresponds to odd indexed terms. - _Rémy Sigrist_, Jan 31 2018 %H A298990 Rémy Sigrist, Table of n, a(n) for n = 0..10000 %H A298990 Rémy Sigrist, C++ program for A298990 %e A298990 2* 2 = 2 + 2, so a(1) = 2. %e A298990 2* 4 = 3 + 5 = 5 + 3, so a(2) = 4. %e A298990 2* 5 = 3 + 7 = 5 + 5 = 7 + 3, so a(3) = 5. %e A298990 2* 8 = 3 + 13 = 5 + 11 = 11 + 5 = 13 + 3, so a(4) = 8. %e A298990 2*11 = 3 + 19 = 5 + 17 = 11 + 11 = 17 + 5 = 19 + 3, so a(5) = 11. %o A298990 (C++) See Links section. %o A298990 (PARI) nb(n) = sum(i=1, 2*n-1, isprime(i) && isprime(2*n-i)); %o A298990 a(n) = {my(k=1); while (nb(k) != n, k++); k;} \\ _Michel Marcus_, Jan 31 2018 %Y A298990 Cf. A035026. %K A298990 nonn,new %O A298990 0,2 %A A298990 _Seiichi Manyama_, Jan 31 2018 %I A298942 %S A298942 12,36,54,84,98,162,242,338,484,578,722,1058,1682,1922,2738,3362,3698, %T A298942 4418,5618,6962,8978,10082,12482,13778,15842,18818,20402,21218,22898, %U A298942 25538,29282,32258,34322,37538,38642,44402,45602,49298,53138,55778,59858,64082,72962 %N A298942 Where records occur in A070138. %Y A298942 Cf. A070138. %K A298942 nonn,new %O A298942 1,1 %A A298942 _Peter Kagey_, Jan 30 2018 %E A298942 a(27)-a(43) from _Giovanni Resta_, Jan 31 2018 %I A299015 %S A299015 1,2,2,4,7,4,8,13,13,8,16,29,20,29,16,32,73,44,44,73,32,64,157,123, %T A299015 174,123,157,64,128,353,343,1052,1052,343,353,128,256,869,957,4488, %U A299015 6908,4488,957,869,256,512,1993,2710,18758,39124,39124,18758,2710,1993,512,1024 %N A299015 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299015 Table starts %C A299015 ...1....2....4......8.......16........32..........64..........128 %C A299015 ...2....7...13.....29.......73.......157.........353..........869 %C A299015 ...4...13...20.....44......123.......343.........957.........2710 %C A299015 ...8...29...44....174.....1052......4488.......18758........89713 %C A299015 ..16...73..123...1052.....6908.....39124......259556......1718835 %C A299015 ..32..157..343...4488....39124....379236.....3848010.....39235328 %C A299015 ..64..353..957..18758...259556...3848010....59666756....931277377 %C A299015 .128..869.2710..89713..1718835..39235328...931277377..22388413097 %C A299015 .256.1993.7749.409166.11081989.404291236.14739630633.545009501463 %H A299015 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299015 Empirical for column k: %F A299015 k=1: a(n) = 2*a(n-1) %F A299015 k=2: a(n) = 4*a(n-1) -5*a(n-2) +10*a(n-3) -24*a(n-4) +16*a(n-5) for n>6 %F A299015 k=3: [order 16] for n>17 %F A299015 k=4: [order 66] for n>70 %e A299015 Some solutions for n=5 k=4 %e A299015 ..0..1..0..1. .0..1..1..0. .0..1..0..1. .0..1..0..1. .0..1..0..0 %e A299015 ..0..0..1..0. .1..1..1..1. .0..0..0..1. .0..1..1..0. .1..1..0..1 %e A299015 ..0..0..1..1. .0..0..1..1. .0..0..1..0. .1..1..1..1. .0..0..0..1 %e A299015 ..1..0..0..0. .1..0..1..1. .1..0..0..1. .0..0..0..0. .1..0..0..0 %e A299015 ..1..0..1..1. .0..0..1..0. .0..1..1..0. .1..1..1..1. .1..0..1..1 %Y A299015 Column 1 is A000079(n-1). %Y A299015 Column 2 is A298215. %K A299015 nonn,tabl,new %O A299015 1,2 %A A299015 _R. H. Hardin_, Jan 31 2018 %I A299014 %S A299014 64,353,957,18758,259556,3848010,59666756,931277377,14739630633, %T A299014 227368007690,3555807950108,55990966099803,872001417883372, %U A299014 13628063102584994,213631343418010157,3338835028830178746 %N A299014 Number of nX7 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299014 Column 7 of A299015. %H A299014 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299014 Some solutions for n=5 %e A299014 ..0..1..0..0..0..0..1. .0..0..1..0..1..0..0. .0..1..1..0..1..0..0 %e A299014 ..0..1..0..1..0..0..0. .0..1..1..0..0..1..1. .0..0..0..0..1..1..1 %e A299014 ..0..1..0..0..1..1..1. .1..1..1..0..0..0..0. .1..1..0..0..0..0..1 %e A299014 ..1..0..0..0..0..0..0. .0..0..1..1..0..0..1. .0..0..1..1..0..0..0 %e A299014 ..0..1..1..0..0..1..0. .1..0..1..0..1..0..1. .0..1..0..1..0..1..1 %Y A299014 Cf. A299015. %K A299014 nonn,new %O A299014 1,1 %A A299014 _R. H. Hardin_, Jan 31 2018 %I A299013 %S A299013 32,157,343,4488,39124,379236,3848010,39235328,404291236,4084330420, %T A299013 41612704929,426841785170,4342251178368,44245587228554, %U A299013 452115024914172,4610407966852939,47006082542714061,479685596824364169 %N A299013 Number of nX6 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299013 Column 6 of A299015. %H A299013 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299013 Some solutions for n=5 %e A299013 ..0..0..1..0..1..1. .0..1..0..1..0..1. .0..1..0..1..0..1. .0..0..0..1..0..1 %e A299013 ..1..1..1..0..1..0. .0..1..1..0..1..1. .1..0..1..0..0..1. .1..1..1..0..0..1 %e A299013 ..0..1..1..1..1..1. .1..1..1..1..1..1. .0..0..0..0..0..1. .0..1..1..0..0..1 %e A299013 ..1..0..1..1..0..0. .0..0..0..1..1..0. .0..0..1..0..0..0. .0..1..1..0..0..1 %e A299013 ..1..0..1..0..1..1. .1..1..1..0..1..0. .0..1..0..1..1..1. .1..0..0..1..0..1 %Y A299013 Cf. A299015. %K A299013 nonn,new %O A299013 1,1 %A A299013 _R. H. Hardin_, Jan 31 2018 %I A299012 %S A299012 16,73,123,1052,6908,39124,259556,1718835,11081989,71873006,470441853, %T A299012 3077586485,20035990710,130783303012,855187074560,5580286107117, %U A299012 36420283697179,237919422581446,1553438331352629,10141105816820335 %N A299012 Number of nX5 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299012 Column 5 of A299015. %H A299012 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299012 Some solutions for n=5 %e A299012 ..0..1..1..1..0. .0..0..1..0..1. .0..1..0..0..1. .0..1..0..0..1 %e A299012 ..1..0..0..0..1. .1..0..1..0..0. .0..1..1..1..1. .0..1..1..1..0 %e A299012 ..0..0..0..0..0. .0..0..0..0..0. .1..0..1..0..0. .1..0..1..0..1 %e A299012 ..1..1..1..0..0. .0..0..1..0..1. .0..1..1..1..1. .0..1..1..1..0 %e A299012 ..0..1..1..1..0. .1..0..1..0..0. .1..0..1..1..0. .1..0..1..1..1 %Y A299012 Cf. A299015. %K A299012 nonn,new %O A299012 1,1 %A A299012 _R. H. Hardin_, Jan 31 2018 %I A299011 %S A299011 8,29,44,174,1052,4488,18758,89713,409166,1788003,8121661,37033578, %T A299011 165692402,746131075,3379582490,15225588917,68575462952,309641196485, %U A299011 1396795549454,6296528560960,28404446913096,128136379596690 %N A299011 Number of nX4 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299011 Column 4 of A299015. %H A299011 R. H. Hardin, Table of n, a(n) for n = 1..210 %H A299011 R. H. Hardin, Empirical recurrence of order 66 %F A299011 Empirical recurrence of order 66 (see link above) %e A299011 Some solutions for n=5 %e A299011 ..0..0..1..1. .0..1..1..0. .0..0..1..0. .0..0..0..1. .0..1..0..0 %e A299011 ..1..1..0..0. .1..0..0..1. .1..1..1..0. .1..0..0..0. .0..0..1..1 %e A299011 ..0..0..0..1. .1..0..0..1. .1..0..1..1. .1..1..1..1. .0..0..0..0 %e A299011 ..1..0..0..0. .0..1..1..0. .1..1..0..0. .1..1..0..1. .1..0..1..0 %e A299011 ..1..0..1..1. .1..1..1..1. .0..0..1..0. .0..1..0..0. .1..0..1..1 %Y A299011 Cf. A299015. %K A299011 nonn,new %O A299011 1,1 %A A299011 _R. H. Hardin_, Jan 31 2018 %I A299010 %S A299010 4,13,20,44,123,343,957,2710,7749,22170,63434,181941,521609,1495695, %T A299010 4290128,12304541,35291808,101228002,290349737,832808429,2388749945, %U A299010 6851656946,19652642153,56369797674,161685824712,463764471339 %N A299010 Number of nX3 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299010 Column 3 of A299015. %H A299010 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299010 Empirical: a(n) = 4*a(n-1) -3*a(n-2) +5*a(n-3) -22*a(n-4) +14*a(n-5) -a(n-6) +24*a(n-7) -a(n-8) -38*a(n-9) +61*a(n-10) -84*a(n-11) +77*a(n-12) -89*a(n-13) +54*a(n-14) -20*a(n-15) +24*a(n-16) for n>17 %e A299010 Some solutions for n=5 %e A299010 ..0..1..0. .0..1..0. .0..1..0. .0..0..1. .0..0..0. .0..1..0. .0..0..1 %e A299010 ..1..0..0. .0..1..0. .1..0..0. .1..1..0. .0..1..0. .1..0..0. .1..0..1 %e A299010 ..1..0..0. .0..1..0. .0..0..0. .0..1..1. .0..0..0. .0..0..0. .0..0..1 %e A299010 ..1..0..1. .0..1..0. .0..0..1. .1..1..1. .1..1..1. .0..0..1. .0..0..0 %e A299010 ..1..0..0. .0..1..0. .0..1..0. .0..0..1. .1..0..1. .1..0..1. .0..1..1 %Y A299010 Cf. A299015. %K A299010 nonn,new %O A299010 1,1 %A A299010 _R. H. Hardin_, Jan 31 2018 %I A299009 %S A299009 1,7,20,174,6908,379236,59666756,22388413097,20692923925827 %N A299009 Number of nXn 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299009 Diagonal of A299015. %e A299009 Some solutions for n=5 %e A299009 ..0..1..0..1..0. .0..0..0..1..0. .0..1..1..0..0. .0..1..0..1..0 %e A299009 ..1..1..0..1..0. .1..0..0..0..1. .1..1..1..1..1. .1..1..0..0..1 %e A299009 ..0..0..0..0..1. .1..0..1..0..1. .0..0..0..1..0. .0..0..0..0..1 %e A299009 ..1..1..0..0..0. .1..0..0..1..0. .1..0..0..0..1. .1..0..1..1..0 %e A299009 ..0..1..0..1..1. .0..1..0..1..1. .1..0..1..1..0. .0..0..1..0..1 %Y A299009 Cf. A299015. %K A299009 nonn,new %O A299009 1,2 %A A299009 _R. H. Hardin_, Jan 31 2018 %I A299008 %S A299008 1,2,2,4,8,4,8,26,26,8,16,88,94,88,16,32,298,372,372,298,32,64,1012, %T A299008 1510,1977,1510,1012,64,128,3440,6105,11553,11553,6105,3440,128,256, %U A299008 11700,24546,63472,111695,63472,24546,11700,256,512,39804,98995,350339,881525 %N A299008 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299008 Table starts %C A299008 ...1.....2......4........8........16..........32...........64............128 %C A299008 ...2.....8.....26.......88.......298........1012.........3440..........11700 %C A299008 ...4....26.....94......372......1510........6105........24546..........98995 %C A299008 ...8....88....372.....1977.....11553.......63472.......350339........1960512 %C A299008 ..16...298...1510....11553....111695......881525......7133441.......61902351 %C A299008 ..32..1012...6105....63472....881525.....9369586....103514554.....1240234582 %C A299008 ..64..3440..24546...350339...7133441...103514554...1591404390....26808046076 %C A299008 .128.11700..98995..1960512..61902351..1240234582..26808046076...667156890263 %C A299008 .256.39804.399424.10931666.518059406.14225843633.428444704211.15251414560006 %H A299008 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299008 Empirical for column k: %F A299008 k=1: a(n) = 2*a(n-1) %F A299008 k=2: a(n) = 4*a(n-1) -2*a(n-2) +2*a(n-3) -6*a(n-4) -4*a(n-5) %F A299008 k=3: [order 18] for n>19 %F A299008 k=4: [order 65] for n>66 %e A299008 Some solutions for n=5 k=4 %e A299008 ..0..1..1..0. .0..0..1..0. .0..0..1..1. .0..1..1..1. .0..0..1..0 %e A299008 ..0..0..1..0. .1..0..1..0. .0..1..0..1. .1..0..0..0. .1..0..0..1 %e A299008 ..1..1..1..1. .0..1..1..0. .0..1..0..1. .1..1..1..1. .0..0..0..1 %e A299008 ..0..0..0..0. .1..1..0..0. .1..0..1..1. .0..0..0..1. .1..0..0..1 %e A299008 ..1..1..1..1. .1..0..1..1. .0..1..1..0. .0..0..0..1. .1..0..0..1 %Y A299008 Column 1 is A000079(n-1). %Y A299008 Column 2 is A298189. %K A299008 nonn,tabl,new %O A299008 1,2 %A A299008 _R. H. Hardin_, Jan 31 2018 %I A299007 %S A299007 64,3440,24546,350339,7133441,103514554,1591404390,26808046076, %T A299007 428444704211,6854463412599,111650146941934,1805590344686823, %U A299007 29161058990902557,472506760744960516,7649634809894141592 %N A299007 Number of nX7 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299007 Column 7 of A299008. %H A299007 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299007 Some solutions for n=5 %e A299007 ..0..1..0..1..0..1..1. .0..1..0..0..0..1..1. .0..1..1..0..0..0..1 %e A299007 ..0..1..0..1..0..1..0. .0..1..0..1..1..1..0. .0..1..0..1..0..1..0 %e A299007 ..0..0..0..1..0..1..0. .0..1..0..1..1..0..1. .0..0..1..1..0..0..1 %e A299007 ..0..1..1..1..0..0..1. .0..1..0..1..1..0..0. .0..1..0..1..0..1..0 %e A299007 ..0..1..0..1..0..1..0. .0..0..1..0..1..1..0. .0..1..0..0..1..1..0 %Y A299007 Cf. A299008. %K A299007 nonn,new %O A299007 1,1 %A A299007 _R. H. Hardin_, Jan 31 2018 %I A299006 %S A299006 32,1012,6105,63472,881525,9369586,103514554,1240234582,14225843633, %T A299006 162650777241,1892744068404,21899542776792,252729477468682, %U A299006 2926551718265169,33872636295144537,391738134300447708 %N A299006 Number of nX6 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299006 Column 6 of A299008. %H A299006 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299006 Some solutions for n=5 %e A299006 ..0..0..0..1..1..1. .0..1..0..1..1..1. .0..0..1..1..1..0. .0..1..1..1..1..1 %e A299006 ..1..1..1..1..0..0. .0..1..1..0..1..0. .1..0..1..0..0..1. .0..0..0..0..0..0 %e A299006 ..0..0..1..1..1..1. .0..1..0..1..1..1. .0..1..1..1..1..1. .0..1..0..1..1..1 %e A299006 ..1..0..0..1..0..1. .0..0..1..1..1..0. .0..1..1..1..0..1. .0..1..1..1..0..0 %e A299006 ..0..0..1..1..0..1. .0..1..0..1..0..1. .0..0..0..1..0..0. .0..0..0..1..1..0 %Y A299006 Cf. A299008. %K A299006 nonn,new %O A299006 1,1 %A A299006 _R. H. Hardin_, Jan 31 2018 %I A299005 %S A299005 16,298,1510,11553,111695,881525,7133441,61902351,518059406, %T A299005 4315727638,36550245275,307995088326,2587830346920,21817284579561, %U A299005 183865862731862,1548119189613657,13042798674877022,109892370175750557 %N A299005 Number of nX5 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299005 Column 5 of A299008. %H A299005 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299005 Some solutions for n=5 %e A299005 ..0..1..0..1..1. .0..1..0..0..1. .0..0..1..1..1. .0..1..0..1..1 %e A299005 ..1..0..0..0..0. .1..0..1..0..0. .0..0..1..0..0. .1..0..0..0..0 %e A299005 ..1..0..0..1..0. .1..1..0..0..1. .0..0..0..0..1. .0..0..0..1..0 %e A299005 ..0..0..0..1..1. .1..0..1..0..0. .0..0..1..1..1. .1..1..0..1..0 %e A299005 ..0..1..0..0..0. .1..0..0..1..0. .1..1..0..0..0. .1..1..0..1..1 %Y A299005 Cf. A299008. %K A299005 nonn,new %O A299005 1,1 %A A299005 _R. H. Hardin_, Jan 31 2018 %I A299004 %S A299004 8,88,372,1977,11553,63472,350339,1960512,10931666,60915771,339863732, %T A299004 1896117726,10577509015,59011882438,329233976176,1836824343961, %U A299004 10247854122236,57174149141128,318982362376709,1779646767539938 %N A299004 Number of nX4 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299004 Column 4 of A299008. %H A299004 R. H. Hardin, Table of n, a(n) for n = 1..210 %H A299004 R. H. Hardin, Empirical recurrence of order 65 %F A299004 Empirical recurrence of order 65 (see link above) %e A299004 Some solutions for n=5 %e A299004 ..0..1..0..1. .0..0..1..0. .0..0..0..1. .0..0..0..0. .0..1..1..1 %e A299004 ..0..1..1..1. .0..1..0..0. .1..1..0..0. .1..1..0..1. .1..0..0..1 %e A299004 ..0..0..0..1. .0..1..1..1. .1..0..0..1. .0..0..0..1. .0..1..1..0 %e A299004 ..0..1..0..1. .0..0..0..0. .0..0..0..0. .1..1..1..0. .1..0..0..1 %e A299004 ..1..1..1..0. .1..1..1..1. .0..1..1..0. .0..1..0..0. .1..1..0..1 %Y A299004 Cf. A299008. %K A299004 nonn,new %O A299004 1,1 %A A299004 _R. H. Hardin_, Jan 31 2018 %I A299003 %S A299003 4,26,94,372,1510,6105,24546,98995,399424,1610936,6496983,26205472, %T A299003 105698076,426321495,1719528280,6935574134,27974034790,112830828296, %U A299003 455093402265,1835579965589,7403653180460,29861995650489,120445781051920 %N A299003 Number of nX3 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299003 Column 3 of A299008. %H A299003 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A299003 Empirical: a(n) = 5*a(n-1) -4*a(n-2) +5*a(n-3) -16*a(n-4) -19*a(n-5) +35*a(n-6) -5*a(n-7) +54*a(n-8) -94*a(n-9) +3*a(n-11) +74*a(n-12) +33*a(n-13) -21*a(n-14) -17*a(n-15) -8*a(n-16) -11*a(n-17) -4*a(n-18) for n>19 %e A299003 Some solutions for n=7 %e A299003 ..0..1..1. .0..0..1. .0..0..0. .0..1..0. .0..0..1. .0..0..0. .0..1..0 %e A299003 ..0..1..0. .1..1..1. .1..1..1. .1..0..1. .1..0..1. .0..1..1. .1..0..1 %e A299003 ..0..1..0. .1..0..1. .0..0..1. .0..1..1. .1..0..1. .1..0..0. .0..0..0 %e A299003 ..1..0..0. .1..0..1. .0..1..0. .0..0..0. .0..0..1. .0..1..1. .1..1..1 %e A299003 ..0..1..0. .0..1..0. .1..1..0. .1..0..0. .1..0..1. .1..0..0. .0..0..0 %e A299003 ..0..1..1. .0..0..0. .1..0..0. .1..0..0. .0..1..1. .0..1..1. .0..0..0 %e A299003 ..0..1..0. .0..1..1. .0..1..1. .1..1..1. .1..0..1. .0..1..0. .1..1..1 %Y A299003 Cf. A299008. %K A299003 nonn,new %O A299003 1,1 %A A299003 _R. H. Hardin_, Jan 31 2018 %I A299002 %S A299002 1,8,94,1977,111695,9369586,1591404390,667156890263,490770543899625 %N A299002 Number of nXn 0..1 arrays with every element equal to 0, 1, 2, 3, 5 or 6 king-move adjacent elements, with upper left element zero. %C A299002 Diagonal of A299008. %e A299002 Some solutions for n=5 %e A299002 ..0..0..1..1..0. .0..1..0..0..0. .0..0..1..1..1. .0..0..1..1..0 %e A299002 ..1..0..1..0..0. .1..0..1..1..0. .1..0..0..0..1. .1..0..1..0..0 %e A299002 ..0..0..0..1..1. .1..1..1..1..0. .1..1..1..1..0. .0..1..1..1..0 %e A299002 ..1..0..0..0..1. .0..1..0..1..1. .0..0..0..0..0. .1..1..0..1..0 %e A299002 ..1..0..1..1..0. .1..0..1..0..1. .1..1..0..1..0. .0..0..0..0..1 %Y A299002 Cf. A299008. %K A299002 nonn,new %O A299002 1,2 %A A299002 _R. H. Hardin_, Jan 31 2018 %I A299001 %S A299001 1,2,2,4,8,4,8,31,31,8,16,121,179,121,16,32,472,1073,1073,472,32,64, %T A299001 1841,6479,10150,6479,1841,64,128,7181,39015,97462,97462,39015,7181, %U A299001 128,256,28010,235033,932318,1502511,932318,235033,28010,256,512,109255,1416220 %N A299001 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 6 king-move adjacent elements, with upper left element zero. %C A299001 Table starts %C A299001 ...1......2.......4.........8..........16............32..............64 %C A299001 ...2......8......31.......121.........472..........1841............7181 %C A299001 ...4.....31.....179......1073........6479.........39015..........235033 %C A299001 ...8....121....1073.....10150.......97462........932318.........8918662 %C A299001 ..16....472....6479.....97462.....1502511......23031390.......353088569 %C A299001 ..32...1841...39015....932318....23031390.....564821426.....13849577141 %C A299001 ..64...7181..235033...8918662...353088569...13849577141....543022804167 %C A299001 .128..28010.1416220..85379274..5420414253..340272636359..21348528789475 %C A299001 .256.109255.8533123.817325435.83214520668.8360888694306.839394891016021 %H A299001 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A299001 Empirical for column k: %F A299001 k=1: a(n) = 2*a(n-1) %F A299001 k=2: a(n) = 3*a(n-1) +3*a(n-2) +2*a(n-3) %F A299001 k=3: [order 10] for n>11 %F A299001 k=4: [order 37] for n>38 %e A299001 Some solutions for n=5 k=4 %e A299001 ..0..1..1..0. .0..0..1..0. .0..0..1..0. .0..1..1..0. .0..1..1..0 %e A299001 ..1..1..1..0. .1..1..0..0. .1..1..0..1. .0..1..0..1. .1..1..1..0 %e A299001 ..1..1..0..0. .1..1..0..1. .0..1..0..1. .1..1..0..1. .0..0..0..0 %e A299001 ..0..0..0..1. .1..0..0..1. .0..0..0..1. .0..0..1..0. .1..1..0..1 %e A299001 ..1..1..1..1. .1..0..0..1. .1..0..0..1. .1..0..0..0. .1..1..0..1 %Y A299001 Column 1 is A000079(n-1). %Y A299001 Column 2 is A281831. %K A299001 nonn,tabl,new %O A299001 1,2 %A A299001 _R. H. Hardin_, Jan 31 2018 %I A299000 %S A299000 64,7181,235033,8918662,353088569,13849577141,543022804167, %T A299000 21348528789475,839394891016021,33001543646989205,1297641776166140920, %U A299000 51026070057748163184,2006456445550940745017,78898675195870846488059 %N A299000 Number of nX7 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 6 king-move adjacent elements, with upper left element zero. %C A299000 Column 7 of A299001. %H A299000 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A299000 Some solutions for n=5 %e A299000 ..0..0..1..1..0..1..0. .0..0..1..0..0..0..1. .0..0..1..1..0..1..0 %e A299000 ..0..1..1..0..1..1..0. .0..1..1..0..1..1..0. .0..1..1..0..1..0..0 %e A299000 ..0..0..0..1..0..0..1. .0..0..0..1..0..0..0. .0..0..0..1..0..1..1 %e A299000 ..0..0..0..1..0..1..0. .0..0..0..1..0..0..0. .0..0..0..1..0..1..1 %e A299000 ..0..1..1..1..1..0..0. .0..1..1..1..0..1..1. .0..1..1..1..0..0..0 %Y A299000 Cf. A299001. %K A299000 nonn,new %O A299000 1,1 %A A299000 _R. H. Hardin_, Jan 31 2018 %I A298999 %S A298999 32,1841,39015,932318,23031390,564821426,13849577141,340272636359, %T A298999 8360888694306,205429207630824,5047897978832238,124042608868375165, %U A298999 3048119277135250592,74902250221349890637,1840597493803525251058 %N A298999 Number of nX6 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 6 king-move adjacent elements, with upper left element zero. %C A298999 Column 6 of A299001. %H A298999 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298999 Some solutions for n=5 %e A298999 ..0..1..0..1..1..0. .0..0..1..0..1..1. .0..1..0..0..1..1. .0..1..0..1..0..1 %e A298999 ..0..0..0..1..0..0. .0..0..0..1..0..0. .0..0..1..1..0..1. .0..0..0..1..0..1 %e A298999 ..0..1..0..1..1..1. .0..1..0..1..1..1. .0..0..0..1..0..1. .0..0..1..1..0..0 %e A298999 ..0..0..1..1..0..0. .0..1..1..0..0..0. .0..1..0..1..1..0. .0..1..0..1..1..1 %e A298999 ..0..0..0..0..0..1. .0..1..1..1..1..0. .0..1..0..1..0..0. .0..0..0..0..0..0 %Y A298999 Cf. A299001. %K A298999 nonn,new %O A298999 1,1 %A A298999 _R. H. Hardin_, Jan 31 2018 %I A298998 %S A298998 16,472,6479,97462,1502511,23031390,353088569,5420414253,83214520668, %T A298998 1277503835363,19613353956412,301126423566543,4623243598098878, %U A298998 70981647492926722,1089798075952103889,16731935000766925329 %N A298998 Number of nX5 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 6 king-move adjacent elements, with upper left element zero. %C A298998 Column 5 of A299001. %H A298998 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298998 Some solutions for n=5 %e A298998 ..0..0..0..0..0. .0..0..0..0..1. .0..0..0..0..0. .0..0..0..0..0 %e A298998 ..1..1..1..1..0. .1..0..0..1..0. .0..1..0..1..0. .1..1..0..0..1 %e A298998 ..0..1..1..0..1. .0..0..1..0..1. .1..1..1..0..1. .1..0..1..0..0 %e A298998 ..1..0..1..1..0. .0..1..1..1..1. .0..0..1..0..0. .0..0..1..0..1 %e A298998 ..0..0..1..0..0. .0..1..0..0..0. .0..1..0..0..1. .0..1..0..0..1 %Y A298998 Cf. A299001. %K A298998 nonn,new %O A298998 1,1 %A A298998 _R. H. Hardin_, Jan 31 2018 %I A298997 %S A298997 8,121,1073,10150,97462,932318,8918662,85379274,817325435,7824101240, %T A298997 74900669973,717032783698,6864239941309,65712258531277, %U A298997 629072225933512,6022192925933026,57651265263937577,551903354684601177 %N A298997 Number of nX4 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 6 king-move adjacent elements, with upper left element zero. %C A298997 Column 4 of A299001. %H A298997 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298997 Empirical: a(n) = 9*a(n-1) +8*a(n-2) +38*a(n-3) -490*a(n-4) -1002*a(n-5) -1061*a(n-6) +8079*a(n-7) +23108*a(n-8) +19262*a(n-9) -30209*a(n-10) -172074*a(n-11) -127690*a(n-12) -120925*a(n-13) +524664*a(n-14) +337275*a(n-15) +702573*a(n-16) -827151*a(n-17) -958523*a(n-18) -1581927*a(n-19) -619058*a(n-20) +414995*a(n-21) +389515*a(n-22) +901857*a(n-23) -168659*a(n-24) +31470*a(n-25) -151466*a(n-26) -27657*a(n-27) +129562*a(n-28) -14754*a(n-29) +50563*a(n-30) -12466*a(n-31) -14531*a(n-32) +10488*a(n-33) -5321*a(n-34) +799*a(n-35) -120*a(n-36) +12*a(n-37) for n>38 %e A298997 Some solutions for n=5 %e A298997 ..0..0..1..1. .0..0..1..0. .0..1..1..1. .0..1..0..1. .0..0..0..1 %e A298997 ..0..0..0..1. .1..1..1..0. .1..1..0..1. .1..0..0..1. .1..0..1..1 %e A298997 ..0..1..0..1. .1..0..0..1. .0..1..0..1. .1..0..1..0. .1..0..1..0 %e A298997 ..0..1..1..0. .1..0..1..1. .0..1..0..0. .0..1..1..0. .1..1..0..1 %e A298997 ..0..0..1..0. .0..1..1..0. .0..0..1..0. .1..0..0..0. .0..0..1..0 %Y A298997 Cf. A299001. %K A298997 nonn,new %O A298997 1,1 %A A298997 _R. H. Hardin_, Jan 31 2018 %I A298996 %S A298996 4,31,179,1073,6479,39015,235033,1416220,8533123,51414948,309794527, %T A298996 1866627347,11247127125,67768156738,408328539256,2460332478992, %U A298996 14824425340207,89322718995185,538202860973128,3242873961665678 %N A298996 Number of nX3 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 6 king-move adjacent elements, with upper left element zero. %C A298996 Column 3 of A299001. %H A298996 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298996 Empirical: a(n) = 5*a(n-1) +6*a(n-2) +7*a(n-3) -26*a(n-4) -53*a(n-5) -27*a(n-6) -39*a(n-7) +17*a(n-8) -5*a(n-9) +12*a(n-10) for n>11 %e A298996 Some solutions for n=6 %e A298996 ..0..0..1. .0..1..1. .0..0..0. .0..0..1. .0..0..1. .0..1..1. .0..1..1 %e A298996 ..1..1..0. .1..0..1. .1..1..0. .1..1..1. .0..1..0. .0..0..0. .0..1..0 %e A298996 ..0..1..0. .0..0..0. .0..0..1. .0..0..0. .1..0..0. .1..0..1. .0..0..0 %e A298996 ..0..1..0. .1..1..1. .0..1..0. .0..1..0. .0..1..1. .0..1..1. .0..0..1 %e A298996 ..0..1..0. .0..0..0. .1..1..1. .1..1..1. .0..1..0. .0..0..0. .0..1..0 %e A298996 ..1..1..0. .0..1..1. .0..1..0. .0..1..0. .0..0..1. .1..1..1. .1..1..0 %Y A298996 Cf. A299001. %K A298996 nonn,new %O A298996 1,1 %A A298996 _R. H. Hardin_, Jan 31 2018 %I A298995 %S A298995 1,8,179,10150,1502511,564821426,543022804167,1343983807513934, %T A298995 8532605215407361244 %N A298995 Number of nXn 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 6 king-move adjacent elements, with upper left element zero. %C A298995 Diagonal of A299001. %e A298995 Some solutions for n=5 %e A298995 ..0..0..0..0..0. .0..0..0..0..1. .0..0..0..0..0. .0..0..0..0..0 %e A298995 ..1..0..1..1..1. .1..0..0..1..0. .1..0..1..1..0. .0..1..1..1..0 %e A298995 ..1..1..0..0..0. .0..0..1..0..0. .1..0..1..0..1. .0..0..1..0..0 %e A298995 ..1..1..1..0..1. .0..1..1..0..1. .0..1..0..1..1. .1..1..1..1..1 %e A298995 ..0..1..1..0..1. .1..0..0..0..1. .1..1..0..1..0. .1..0..0..1..0 %Y A298995 Cf. A299001. %K A298995 nonn,new %O A298995 1,2 %A A298995 _R. H. Hardin_, Jan 31 2018 %I A298411 %S A298411 1,-2,-10,-20,-90,132,-836,6040,2310,60180,180308,1662568,-2995620, %T A298411 24401320,44072120,-102437328,19390406,2649221300,-10584460060, %U A298411 14475802440,-228570333836,-815899620616,2088529753800,-5590702681520,-100828534100580,-172013432412024 %N A298411 Coefficients of q^(-1/24)*eta(4q)^(1/2). %C A298411 The q^(kn) term of any single factor of the product (1-(4q)^k)^(1/2) is (-2)*A000108(n-1). Hence these numbers are related to the Catalan numbers A000108 by a partition-based convolution. %C A298411 Sequence appears to be positive and negative roughly half the time. %H A298411 Seiichi Manyama, Table of n, a(n) for n = 0..1000 %F A298411 G.f.: Product_{k>=1} (1 - (4x)^k)^(1/2). %t A298411 Series[Product[(1 - (4 q)^k)^(1/2), {k, 1, 100}], {q, 0, 100}] %Y A298411 Cf. A000108, A271235. %K A298411 sign,easy,new %O A298411 0,2 %A A298411 _William J. Keith_, Jan 18 2018 %I A298702 %S A298702 1,2,3,6,11,15,17,19,22,51 %N A298702 Numbers k such that k!+1 reversed is a prime. %t A298702 Do[If[PrimeQ[FromDigits[Reverse[IntegerDigits[n! + 1]]]], Print[n]], {n, 400}] (* _Vincenzo Librandi_, Jan 25 2018 *) %o A298702 (PARI) isok(n) = isprime(fromdigits(Vecrev(digits(n!+1)))); \\ _Michel Marcus_, Jan 26 2018 %Y A298702 Cf. A002981. %K A298702 nonn,base,more,new %O A298702 1,2 %A A298702 _Paolo Galliani_, Jan 25 2018 %E A298702 More terms from _Vincenzo Librandi_, Jan 25 2018 %I A297969 %S A297969 1,3,4,2,5,7,8,6,9,11,13,30,21,15,17,19,31,33,35,37,39,51,34,23,36,25, %T A297969 50,20,27,38,29,70,22,24,41,52,26,43,40,56,53,45,65,28,42,55,60,44,57, %U A297969 46,59,48,73,62,71,75,61,58,77,47,49,64,74,78,63,67,81,72,79,90,91,92,68,95,69,93 %N A297969 Pick any digit "d" in the sequence; the sum ["d" + (the next d digits)] is always even. %C A297969 The sequence starts with a(1) = 1 and was always extended with the smallest available integer not yet present in the sequence. This is the lexicographically first sequence with this property. %H A297969 Lars Blomberg, Table of n, a(n) for n = 1..10000 %e A297969 1 + 3 (the next digit) = 4 (even); %e A297969 3 + 4,2,5 (the next 3 digits) = 14 (even); %e A297969 4 + 2,5,7,8 (the next 4 digits) = 26 (even); %e A297969 2 + 5,7 (the next 2 digits) = 14 (even); %e A297969 ... %e A297969 9 + 1,1,1,3,3,0,2,1,1 = 22 (even); %e A297969 ... %e A297969 0 + (no digit) = 0 (even). %K A297969 base,nonn,new %O A297969 1,2 %A A297969 _Eric Angelini_ and _Lars Blomberg_, Jan 10 2018 %I A297627 %S A297627 52,152,1052,1152,2152,2513,3152,4152,4316,5152,5201,5212,6152,6213, %T A297627 7152,8152,9152,10152,11052,11152,12152,12513,13152,14152,14316,15152, %U A297627 15201,15212,16152,16213,17152,18152,19152,20521,21052,21152,25103,25113,30251,30621,31052,31152,32519,41052,41152,43106 %N A297627 Anagrexpo integers: integers N that exactly reproduce their set of digits when we form the set of exponentiation of pairs of adjacent digits, from left to right. %C A297627 The sequence is infinite, since any term of the sequence can be preceded by as many 1s as needed. The name "anagrexpo integers" comes from "anagram by exponentiation". The same idea is explored by the "anagraprod integers" and the "anagrasum integers" (see "Crossrefs" section hereunder). %H A297627 Jean-Marc Falcoz, Table of n, a(n) for n = 1..7707 %e A297627 a(2) = 152 reproduces the digits 1, 5 and 2 (in a different order) when the exponentiations 1^5=1 and 5^2=25 are taken. The same with a(6) = 2513, which reproduces the digits 2, 5, 1, and 3 when the exponentiations 2^5=32, 5^1=5 and 1^3=1 are taken. %t A297627 Unprotect[Power]; Power[0, 0] := 1; Protect[Power]; Select[Range[10^5], SameQ @@ {Sort@ Flatten@ Map[IntegerDigits[Power @@ #] &, Partition[#, 2, 1]], Sort@ #} &@ IntegerDigits@ # &] (* _Michael De Vlieger_, Jan 02 2018 *) %Y A297627 Cf. A296451, A296521. %K A297627 base,nonn,new %O A297627 1,1 %A A297627 _Eric Angelini_ and _Jean-Marc Falcoz_, Jan 02 2018 %I A298699 %S A298699 419,5039,51239,513239,5133239,51333239,5133333333333239, %T A298699 513333333333333239,5133333333333333333239,5133333333333333333333239, %U A298699 513333333333333333333333239,513333333333333333333333333333333333333333333333239 %N A298699 Primes of the form A132583(k)*42 - 43. %C A298699 The corresponding values of k are 0, 1, 2, 3, 4, 5, 13, 15, 19, 22, 24, 48, 59, 187, 215, 232. - _Bruno Berselli_, Jan 29 2018 %e A298699 5039 is prime and 5039 = 121*42 - 43, hence it is in the sequence. %e A298699 51239 is prime and 51239 = 1221*42 - 43, hence it is in the sequence. %e A298699 513333239 = 12222221*42 - 43 = 61*1747*4817 is not prime, therefore it is not in the sequence. %t A298699 Select[42 NestList[10 # + 11 &, 11, 50] - 43, PrimeQ] (* _Michael De Vlieger_, Feb 01 2018, after _Harvey P. Dale_ at A132583 *) %Y A298699 Cf. A003020, A132583. %K A298699 nonn,base,less,more,new %O A298699 1,1 %A A298699 _Paolo Galliani_, Jan 27 2018 %E A298699 a(8) - a(12) from _Bruno Berselli_, Jan 29 2018 %I A298613 %S A298613 31,73,157,12763,255127,40952047,524287262143,41943032097151, %T A298613 6871947673534359738367,7036874417766335184372088831, %U A298613 22517998136852471125899906842623,14757395258967641292773786976294838206463,604462909807314587353087302231454903657293676543 %N A298613 Primes formed by the concatenation of 2^k-1 and 2^(k-1)-1. %C A298613 Conjectures: %C A298613 (1) The factorization of a(n) + 1 never contains an odd prime squared. %C A298613 (2) a(n) + 1 is not divisible by 7. %C A298613 (3) k is not congruent to 6 (mod 7). %C A298613 (4) There are infinitely many primes of this form. %p A298613 P:=proc(n) local a; %p A298613 a:=2^(n-1)-1+(2^n-1)*10^(ilog10(2^(n-1)-1)+1); if isprime(a) then a; %p A298613 fi; end: seq(P(i),i=2..10^2); # _Paolo P. Lava_, Jan 23 2018 %t A298613 Select[Map[#1 10^IntegerLength@ #2 + #2 & @@ Reverse@ # &, Partition[Array[2^# - 1 &, 90], 2, 1]], PrimeQ] (* _Michael De Vlieger_, Jan 23 2018 *) %o A298613 (PARI) lista(nn) = for (n=1, nn, if (isprime(p=fromdigits(concat(digits(2^n-1), digits(2^(n-1)-1)))), print1(p, ", "))); \\ _Michel Marcus_, Jan 29 2018 %o A298613 (MAGMA) [t: n in [1..100] | IsPrime(t) where t is Seqint(Intseq(2^(n-1)-1) cat Intseq(2^n-1))]; // _Bruno Berselli_, Feb 02 2018 %Y A298613 Cf. A000040, A000225. %K A298613 nonn,base,new %O A298613 1,1 %A A298613 _Paolo Galliani_, Jan 23 2018 %I A298321 %S A298321 1,1,1,1,2,3,3,4,3,8,6,9,8,9,12,13,11,13,12,16,18,19,18,19,21 %N A298321 Nekrasov-Okounkov sequence. %C A298321 a(n) is the degree in terms of z of the coefficient of x^n's highest degree irreducible factor in Prod_{m>=1} (1-x^m)^(z-1). This can be calculated by reducing the polynomial in the Nekrasov-Okounkov formula. %H A298321 Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, Annales de l'institut Fourier, 60 no. 1 (2010), p. 1-29. %H A298321 Nikita A. Nekrasov and Andrei Okounkov, Seiberg-Witten Theory and Random Partitions, arXiv:hep-th/0306238, 2003. %e A298321 For n = 5, a(n) = 2 because the coefficient of x^5 is Prod_{m>=1} (1-x^m)^(z-1). This can be factorized as -(z-7)(z-4)(z-1)(z^2-23z+30)/120. %K A298321 nonn,more,new %O A298321 1,5 %A A298321 _Kenta Suzuki_, Jan 17 2018 %I A298882 %S A298882 1,2,3,4,5,8,7,16,6,32,11,64,13,128,9,256,17,512,19,1024,12,2048,23, %T A298882 4096,10,8192,18,16384,29,32768,31,65536,24,131072,15,262144,37, %U A298882 524288,27,1048576,41,2097152,43,4194304,36,8388608,47,16777216,14,33554432,48 %N A298882 a(1) = 1, and for any n > 1, if n is the k-th number with least prime factor p, then a(n) is the k-th number with greatest prime factor p. %C A298882 This sequence is a permutation of the natural numbers, with inverse A298268. %C A298882 For any prime p and k > 0: %C A298882 - if s_p(k) is the k-th p-smooth number and r_p(k) is the k-th p-rough number, %C A298882 - then a(p * r_p(k)) = p * s_p(k), %C A298882 - for example: a(11 * A008364(k)) = 11 * A051038(k). %H A298882 Index entries for sequences that are permutations of the natural numbers %F A298882 a(1) = 1. %F A298882 a(A083140(n, k)) = A125624(n, k) for any n > 0 and k > 0. %F A298882 a(n) = A125624(A055396(n), A078898(n)) for any n > 1. %F A298882 Empirically: %F A298882 - a(n) = n iff n belongs to A046022, %F A298882 - a(2 * k) = 2^k for any k > 0, %F A298882 - a(p^2) = 2 * p for any prime p, %F A298882 - a(p * q) = 3 * p for any pair of consecutive odd primes (p, q). %e A298882 The first terms, alongside A020639(n), are: %e A298882 n a(n) lpf(n) %e A298882 -- ---- ------ %e A298882 1 1 1 %e A298882 2 2 2 %e A298882 3 3 3 %e A298882 4 4 2 %e A298882 5 5 5 %e A298882 6 8 2 %e A298882 7 7 7 %e A298882 8 16 2 %e A298882 9 6 3 %e A298882 10 32 2 %e A298882 11 11 11 %e A298882 12 64 2 %e A298882 13 13 13 %e A298882 14 128 2 %e A298882 15 9 3 %e A298882 16 256 2 %e A298882 17 17 17 %e A298882 18 512 2 %e A298882 19 19 19 %e A298882 20 1024 2 %Y A298882 Cf. A008364, A020639, A046022, A051038, A055396, A078898, A083140, A125624, A298268 (inverse). %K A298882 nonn,new %O A298882 1,2 %A A298882 _Rémy Sigrist_, Jan 28 2018 %I A298268 %S A298268 1,2,3,4,5,9,7,6,15,25,11,21,13,49,35,8,17,27,19,55,77,121,23,33,65, %T A298268 169,39,91,29,85,31,10,143,289,119,45,37,361,221,95,41,133,43,187,115, %U A298268 529,47,51,161,125,323,247,53,57,209,203,437,841,59,145,61,961 %N A298268 a(1) = 1, and for any n > 1, if n is the k-th number with greatest prime factor p, then a(n) is the k-th number with least prime factor p. %C A298268 This sequence is a permutation of the natural numbers, with inverse A298882. %C A298268 For any prime p and k > 0: %C A298268 - if s_p(k) is the k-th p-smooth number and r_p(k) is the k-th p-rough number, %C A298268 - then a(p * s_p(k)) = p * r_p(k), %C A298268 - for example: a(11 * A051038(k)) = 11 * A008364(k). %H A298268 Rémy Sigrist, Table of n, a(n) for n = 1..10000 %H A298268 Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (with prime and semiprime values highlighted) %H A298268 Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A006530(n)) %H A298268 Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A176506(k) when a(n) is the k-th semiprime) %H A298268 Rémy Sigrist, PARI program for A298268 %H A298268 Index entries for sequences that are permutations of the natural numbers %F A298268 a(1) = 1. %F A298268 a(A125624(n, k)) = A083140(n, k) for any n > 0 and k > 0. %F A298268 a(n) = A083140(A061395(n), A078899(n)) for any n > 1. %F A298268 Empirically: %F A298268 - a(n) = n iff n belongs to A046022, %F A298268 - a(2^k) = 2 * k for any k > 0, %F A298268 - a(2 * p) = p^2 for any prime p, %F A298268 - a(3 * p) = p * A151800(p) for any odd prime p. %e A298268 The first terms, alongside A006530(n), are: %e A298268 n a(n) gpf(n) %e A298268 -- ---- ------ %e A298268 1 1 1 %e A298268 2 2 2 %e A298268 3 3 3 %e A298268 4 4 2 %e A298268 5 5 5 %e A298268 6 9 3 %e A298268 7 7 7 %e A298268 8 6 2 %e A298268 9 15 3 %e A298268 10 25 5 %e A298268 11 11 11 %e A298268 12 21 3 %e A298268 13 13 13 %e A298268 14 49 7 %e A298268 15 35 5 %e A298268 16 8 2 %e A298268 17 17 17 %e A298268 18 27 3 %e A298268 19 19 19 %e A298268 20 55 5 %o A298268 (PARI) See Links section. %Y A298268 Cf. A006530, A008364, A046022, A051038, A061395, A078899, A083140, A125624, A151800, A176506, A298268, A298882 (inverse). %K A298268 nonn,new %O A298268 1,2 %A A298268 _Rémy Sigrist_, Jan 27 2018 %I A298947 %S A298947 1,1,2,3,6,7,11,12,15,19,22,22,29,32,32,38,42,44,49,51,54,63 %N A298947 Number of integer partitions y of n such that exactly one permutation of y is a Lyndon word. %e A298947 The a(6) = 7 partitions are (6), (51), (42), (411), (3111), (2211), (21111). This list does not include (321) because there are two possible permutations that are Lyndon words, namely (123) and (132). The list does not include (33), (222), or (111111) because no permutation of these is a Lyndon word. %t A298947 LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ]; %t A298947 Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],LyndonQ]]===1&]],{n,20}] %Y A298947 Cf. A000041, A000740, A001037, A032741, A059966, A144300, A167934, A293511, A292444, A298941. %K A298947 nonn,more,new %O A298947 1,3 %A A298947 _Gus Wiseman_, Jan 30 2018 %I A298971 %S A298971 0,1,1,2,1,4,1,5,3,8,1,16,1,20,9,35,1,69,1,110,21,188,1,381,7,632,59, %T A298971 1184,1,2300,1,4115,189,7712,25,14939,1,27596,633,52517,1,101050,1, %U A298971 190748,2247,364724,1,703331,19,1342283,7713,2581430,1,4985609,193 %N A298971 Number of compositions of n that are proper powers of Lyndon words. %C A298971 a(n) is the number of compositions of n that are not Lyndon words but are of the form p * p * ... * p where * is concatenation and p is a Lyndon word. %F A298971 a(n) = Sum_{d|n} (2^d-1)*(phi(n/d)-mu(n/d))/n. %F A298971 a(n) = A008965(n) - A059966(n). %e A298971 The a(12) = 16 compositions: 111111111111, 1111211112, 11131113, 112112112, 11221122, 114114, 12121212, 123123, 131313, 132132, 1515, 222222, 2424, 3333, 444, 66. %t A298971 Table[Sum[DivisorSum[d,MoebiusMu[d/#]*(2^#-1)&]/d,{d,Most@Divisors[n]}],{n,100}] %o A298971 (PARI) a(n) = sumdiv(n, d, (2^d-1)*(eulerphi(n/d)-moebius(n/d))/n); \\ _Michel Marcus_, Jan 31 2018 %Y A298971 Cf. A000005, A000031, A000740, A000961, A001045, A008965, A019536, A034691, A051953, A052823, A059966, A060223, A178472, A185700, A296302, A296373. %K A298971 nonn,new %O A298971 1,4 %A A298971 _Gus Wiseman_, Jan 30 2018 %I A298941 %S A298941 1,1,0,1,1,1,0,0,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,0,1,0,1,1,2,1,0,1,1,1, %T A298941 1,1,1,1,1,1,2,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,3,1,1,1,0,1,2,1,1,1, %U A298941 2,1,2,1,1,1,1,1,2,1,1,0,1,1,3,1,1,1,1,1,3 %N A298941 Number of permutations of the multiset of prime factors of n > 1 that are Lyndon words. %e A298941 The a(90) = 3 Lyndon permutations are {2,3,3,5}, {2,3,5,3}, {2,5,3,3}. %t A298941 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]; %t A298941 LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ]; %t A298941 Table[Length[Select[Permutations[primeMS[n]],LyndonQ]],{n,2,60}] %Y A298941 Cf. A000740, A001037, A001222, A008480, A008965, A059966, A060223, A096443, A112798, A215366, A275024, A294859, A298947. %K A298941 nonn,new %O A298941 2,29 %A A298941 _Gus Wiseman_, Jan 29 2018 %I A296306 %S A296306 1,5,1,21,1,5,1,85,1,5,1,21,1,5,1,341,1,5,1,21,1,5,1,85,1,5,1,21,1,5, %T A296306 1,1365,1,5,1,21,1,5,1,85,1,5,1,21,1,5,1,341,1,5,1,21,1,5,1,85,1,5,1, %U A296306 21,1,5,1,5461,1,5,1,21,1,5,1,85,1,5,1,21,1,5,1,341,1,5,1,21,1,5,1,85,1,5,1,21,1,5,1,1365 %N A296306 a(n) = A001157(n)/A050999(n). %C A296306 a(n) is the sum of the second powers of the divisors of n divided by the sum of the second powers of the odd divisors of n. %C A296306 Conjecture 1: For any nonnegative integer k and positive integer n, the sum of the k-th powers of the divisors of n is divisible by the sum of the k-th powers of the odd divisors of n. %C A296306 Conjecture 2: Distinct terms form A002450 without A002450(0). In other words, a(2^(n-1)) = A002450(n), for n > 0. %C A296306 Conjecture 3: For n > 0, the list of the first 2^n - 1 terms is palindromic. %C A296306 Conjecture 4: For n > 0, the sum of the first 2^n - 1 terms equals A006095(n+1). %F A296306 a(n) = [4^(A007814(n) + 1) - 1]/3. - _David Radcliffe_, Dec 11 2017 %e A296306 A001157(4) = 21 and A050999(4) = 1, therefore a(4) = A001157(4)/A050999(4) = 21. %t A296306 f[n_]:=DivisorSigma[2,n]/Total[Select[Divisors[n],OddQ]^2]; f/@Range[100] %o A296306 (PARI) a(n) = sigma(n, 2)/sumdiv(n, d, d^2*(d % 2)); \\ _Michel Marcus_, Dec 11 2017 %Y A296306 Cf. A001157, A050999. %K A296306 easy,nonn,mult,new %O A296306 1,2 %A A296306 _Ivan N. Ianakiev_, Dec 10 2017 %I A298949 %S A298949 1,-1,0,-1,2,-2,2,-2,2,-3,4,-3,3,-5,5,-5,7,-7,7,-9,10,-11,12,-12,13, %T A298949 -16,18,-17,18,-21,23,-25,26,-27,29,-32,35,-36,37,-40,43,-46,50,-51, %U A298949 52,-58,63,-64,67,-71,73,-79,85,-85,88,-96,100,-104,111,-113,117 %N A298949 Expansion of Product_{k>=2} 1/(1 + x^{F_k}) where F_k are the Fibonacci numbers. %H A298949 Seiichi Manyama, Table of n, a(n) for n = 0..1000 %F A298949 Convolution inverse of A000119. %Y A298949 Cf. A000045, A000119, A003107, A093996. %K A298949 sign,new %O A298949 0,5 %A A298949 _Seiichi Manyama_, Jan 30 2018 %I A298952 %S A298952 1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0,1, %T A298952 1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0,1, %U A298952 0,1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0 %N A298952 Start with a(0) = 1 and add at step n >= 0 the term 1 at position 2*n + a(n). %C A298952 Sum_{i = 0..n} a(i)/n tends to 1/2 as n tends to infinity. [corrected by _Rémy Sigrist_, Jan 31 2018] %e A298952 Set a(n) = 0. %e A298952 n = 0, a(0) = 1. Add term 1 at position 2*0+1 = 1. We have {1,1,0,0,0,0,0,0,0,0,...} %e A298952 n = 1, a(1) = 1. Add term 1 at position 2*1+1 = 3. We have {1,1,0,1,0,0,0,0,0,0,...} %e A298952 n = 2, a(2) = 0. Add term 1 at position 2*2+0 = 4. We have {1,1,0,1,1,0,0,0,0,0,...} %e A298952 n = 3, a(3) = 1. Add term 1 at position 2*3+1 = 7. We have {1,1,0,1,1,0,0,1,0,0,...} %e A298952 and so on. %Y A298952 Cf. A298307. %K A298952 nonn,new %O A298952 0 %A A298952 _Ctibor O. Zizka_, Jan 30 2018 %I A298307 %S A298307 1,1,1,0,2,0,1,0,1,2,0,0,1,1,2,0,0,1,1,0,1,2,0,0,1,2,0,0,1,1,2,0,0,1, %T A298307 1,0,2,0,1,0,1,1,2,0,0,1,2,0,0,1,1,0,2,0,1,0,1,1,2,0,0,1,1,0,2,0,1,0, %U A298307 2,0,1,0,1,2,0,0,1,1,1,0,2,0,1,0,2,0,1,0,1,2,0,0,1,1,2,0,0,1,1,0,2,0,1,0,1,2,0,0,1 %N A298307 Start with a(0) = 1 and add at step n >= 0 the term 1 at position floor(4*(n+a(n))/3). %C A298307 Sum_{i=0..n} a(i)/n = 3/4. For sequences of the type: a(0) = 1, in step n >= 0 add the term 1 at position floor(k*(n+a(n)), k rational number > 1 we have Sum_{i=0..n} a(i)/n = 1/k. %e A298307 Define a sequence b whose terms are initially b(0)=1 and, for n > 0, b(n)=0, i.e., b = {1,0,0,0,0,0,0,0,0,...}; then, for n = 0,1,2,..., increment b(floor(4*(n+b(n))/3)) by 1. For n >= 0, a(n) is the final value of b(n). %e A298307 Sequence b after b(k) is %e A298307 n b(n) k=floor(4*(n+b(n))/3) incremented by 1 %e A298307 = ==== ===================== =============================== %e A298307 {1,0,0,0,0,0,0,0,0,0,0,0,0,...} %e A298307 0 1 floor(4*(0+1)/3) = 1 {1,1,0,0,0,0,0,0,0,0,0,0,0,...} %e A298307 1 1 floor(4*(1+1)/3) = 2 {1,1,1,0,0,0,0,0,0,0,0,0,0,...} %e A298307 2 1 floor(4*(2+1)/3) = 4 {1,1,1,0,1,0,0,0,0,0,0,0,0,...} %e A298307 3 0 floor(4*(3+0)/3) = 4 {1,1,1,0,2,0,0,0,0,0,0,0,0,...} %e A298307 4 2 floor(4*(4+2)/3) = 8 {1,1,1,0,2,0,0,0,1,0,0,0,0,...} %e A298307 5 0 floor(4*(5+0)/3) = 6 {1,1,1,0,2,0,1,0,1,0,0,0,0,...} %e A298307 6 1 floor(4*(6+1)/3) = 9 {1,1,1,0,2,0,1,0,1,1,0,0,0,...} %e A298307 7 0 floor(4*(7+0)/3) = 9 {1,1,1,0,2,0,1,0,1,2,0,0,0,...} %e A298307 8 1 floor(4*(8+1)/3) = 12 {1,1,1,0,2,0,1,0,1,2,0,0,1,...} %t A298307 mx = 104; t = Join[{1}, 0 Range@mx]; k = 1; While[4 k < 3 (mx + 2), t[[ Floor[ 4(k + t[[k]])/3]]]++; k++]; Join[{1}, t] (* _Robert G. Wilson v_, Jan 18 2018 *) %Y A298307 Cf. A136119. %K A298307 nonn,new %O A298307 0,5 %A A298307 _Ctibor O. Zizka_, Jan 16 2018 %I A298982 %S A298982 0,0,1,0,0,4,5,7,0,1,0,0,0,2,0,0,3,0,0,4,0,5,0,0,0,7,0,8,0,9,0,0,11,0, %T A298982 0,13,14,0,0,16,17,18,0,0,0,0,0,0,0,25,0,0,0,0,0,0,33,34,35,36,0,39,4, %U A298982 41,0,44,45,0,48,49,51,52,54,55,0,58,6,61,63,64,66,68,69,71,73,74,76,78,8,81,83,85,87,89,91,93,95,97,0,1 %N A298982 a(n) is the least k for which the most significant decimal digits of k/n (disregarding any leading zeros) are n, or 0 if no such k exists. %C A298982 By decimal digits we mean those of the fractional part of k/n. Otherwise said, we require floor(10^m*k/n) = n for some k < n and m. %C A298982 Indices of 0's are listed in A298981, indices of the other terms are listed in A298980. %C A298982 It appears that the asymptotic density of 0's is below 45%: The number of 0's among a(1..10^k) is (5, 42, 461, 4553, 45423, 451315, 4506142, ...). Is there a simple estimate for the exact value? - _M. F. Hasler_, Feb 01 2018 %H A298982 M. F. Hasler, Table of n, a(n) for n = 1..10000 %e A298982 a(1) = 0 since there does not exist any k such that k/1 has a decimal digit which begins with 1 (cf. comment). %e A298982 a(6) = 4 since 4/6 = 0.666... and its decimal digit begins with 6. %e A298982 a(28) = 8 since 8/28 = 0.28571428571428... even though 1/28 = 0.0357142857142857... has "28" as a subsequence. %t A298982 f[n_] := Compile[{{n, _Integer}}, Block[{k = 1, il = IntegerLength@ n}, While[m = 10^il*k/n; While[ IntegerLength@ Floor@ m < il, m *= 10]; k < n && Floor[m] != n, k++]; If[k < n, k, 0]]]; Array[f, 100] %o A298982 (PARI) A298982(n,k=(n^2-1)\10^(logint(n,10)+1)+1)={k*10^(logint((n^2-(n>1))\k, 10)+1)\n==n && return(k\10^valuation(k,10))} \\ _M. F. Hasler_, Feb 01 2018 %Y A298982 Cf. A298232, A051626, A298980, A298981. %K A298982 easy,nonn,new %O A298982 1,6 %A A298982 _Eric Angelini_, _M. F. Hasler_ and _Robert G. Wilson v_, Jan 30 2018 %I A298981 %S A298981 1,2,4,5,9,11,12,13,15,16,18,19,21,23,24,25,27,29,31,32,34,35,38,39, %T A298981 43,44,45,46,47,48,49,51,52,53,54,55,56,61,65,68,75,99,101,102,103, %U A298981 104,105,106,107,108,109,110,111,112,113,115,116,117,119,120,121,123,124,125,127,128,129,131,132,133,135,136,137,138,139 %N A298981 Numbers n such that there does not exist an integer k < n for which the initial decimal digits of k/n are n. %C A298981 Inspired by A298232. %F A298981 Complement of A298980. %e A298981 2 is in the sequence since there is no k such that k/2 would result in a decimal number which begins with 2, i.e., 0.2000. Instead, the decimal number for odd k's begin with 0.5. %t A298981 fQ = Compile[{{n, _Integer}}, Block[{k = 1, il = IntegerLength@ n}, While[m = 10^il*k/n; While[ IntegerLength@ Floor@ m < il, m *= 10]; k < n && Floor[m] != n, k++]; k == n]]; Select[Range@140, fQ] %o A298981 (PARI) is_A298981(n)=!A298982(n) %Y A298981 Cf. A298232, A051626, A298980, A298982. %K A298981 easy,nonn,new %O A298981 1,2 %A A298981 _Eric Angelini_, _M. F. Hasler_ and _Robert G. Wilson v_, Jan 30 2018 %I A298980 %S A298980 3,6,7,8,10,14,17,20,22,26,28,30,33,36,37,40,41,42,50,57,58,59,60,62, %T A298980 63,64,66,67,69,70,71,72,73,74,76,77,78,79,80,81,82,83,84,85,86,87,88, %U A298980 89,90,91,92,93,94,95,96,97,98,100,114,118,122,126,130,134,141,148,158,161,164,167,170,173,176,184,187 %N A298980 Numbers n such that there exists an integer k < n for which the significant decimal digits of k/n (i.e., neglecting leading zeros) are those of n. %C A298980 Otherwise said, floor(10^m*k/n) = n for some k and m. %C A298980 Also, numbers n which have n as a subsequence in the decimal expansion of k/n, 0 < k < n. %C A298980 Initially it appears that if n is present so is 10n and 11n. These two statements are false. 14 is present but 140 is not. 1/140 = 0.00714285... 17 is present but 187 is not. %C A298980 However if there is a k between 0 and n so that the GCD(k,n) = r > 1 and k/r is used to show that n/r is a member, then so is n. As an example, 33 is a member since 11/33 = 1/3 and 3 is a member. See the first example. %C A298980 The density of numbers in this sequence appears to increase to above 55% near n ~ 10^9. See A298981 for the complement and A298982 for the k-values. %e A298980 3 is a member since 1/3 = 0.3333... and this decimal number begins with 3; %e A298980 6 is a member since 10/6 = 1.666... and its decimal part begins with 6; %e A298980 7 is a member since 5/7 = 0.714285... and its decimal part begins with 7; %e A298980 8 is a member since 7/8 = 0.87500... and its decimal part begins with 8; %e A298980 10 is a member since 1/10 = 0.1000... and its decimal part begins with 10; %e A298980 14 is a member since 2/14 = 0.142857... and its decimal part begins with 14; %e A298980 17 is a member since 3/17 = 0.17647058823 ... and its decimal part begins with 17; etc. %t A298980 fQ = Compile[{{n, _Integer}}, Block[{k = 1, il = IntegerLength@ n}, While[m = 10^il*k/n; While[ IntegerLength@ Floor@ m < il, m *= 10]; k < n && Floor[m] != n, k++]; k < n]]; Select[Range@200, fQ] %o A298980 (PARI) is_A298980(n,k=(n^2-1)\10^(logint(n,10)+1)+1)={k*10^(logint((n^2-(n>1))\k,10)+1)\n==n} \\ Or use A298982 to get the k-value if n is in this sequence or 0 else. \\ _M. F. Hasler_, Feb 01 2018 %Y A298980 Inspired by and equal to the range (= sorted terms) of A298232. %Y A298980 Complement of A298981. %Y A298980 Cf. A051626, A298982. %K A298980 easy,nonn,new %O A298980 1,1 %A A298980 _Eric Angelini_, _M. F. Hasler_ and _Robert G. Wilson v_, Jan 30 2018 %I A298636 %S A298636 1,1,1,1,3,1,1,6,9,1,1,10,36,23,1,1,15,100,181,53,1,1,21,225,845,775, %T A298636 115,1,1,28,441,2890,5957,2956,241,1,1,36,784,8036,30862,36148,10426, %U A298636 495,1,1,45,1296,19278,122276,278530,195934,34899,1005,1,1,55,2025,41406,398874,1560118 %N A298636 Square array T(m,n) = number of ways to draw m-1 horizontal lines [a(i),b(i)] with 0 <= a(i) < b(i) <= n such that if two lines start or end on the same coordinate, no intermediate line crosses this coordinate (see comments); m, n >= 1. %C A298636 Following the OEIS standard, the array is read by falling antidiagonals, i.e., T(1,1), T(1,2), T(2,1), T(1,3), .... %C A298636 "Horizontal line [a(i),b(i)]" means a line from (a(i),i) to (b(i),i). "No intermediate line crosses..." means that, if {a(i),b(i)} and {a(j),b(j)} have x in common for some j > i, then for all i < k < j, either a(k) >= x or b(k) <= x. %C A298636 Equivalently, number of (m-1) X n binary (0,1) matrices where each row has exactly one run of 1's and any two of these runs may not start or end at the same column border, unless no run in the intermediate rows crosses (= extends to both sides of) this border. %C A298636 This construction is relevant for enumerating the tight pavings defined by Knuth in A285357, see his Christmas Tree Lecture video there. %e A298636 The table starts (cf. "table" link): %e A298636 1 1 1 1 1 1 1 ... %e A298636 1 3 6 10 15 21 28 ... (= A000217 = n -> n(n+1)/2) %e A298636 1 9 36 100 225 441 784 ... (= A000537 = A000217^2) %e A298636 1 23 181 845 2890 8036 19278... %e A298636 1 53 775 5957 30862 122276 ... %e A298636 1 115 2956 36148 ... %e A298636 ... %e A298636 Column 2 is A183155. %e A298636 The T(2,3) = 6 drawings are { [0-1], [0-2], [0-3], [1-2], [1-3], [2-3] }. %e A298636 The T(3,2) = 9 drawings are { [0-1; 0-1], [0-1; 0-2], [0-1; 1-2], [0-2; 0-1], [0-2, 0-2], [0-2; 1-2], [1-2; 0-1], [1-2; 0-2], [1-2; 1-2] }. %e A298636 The "no line crosses" condition becomes effective only for m > 3. For m = 4, it excludes drawings like, e.g., [0-1; 0-2; 0-1], [0-1; 0-2; 1-2], ... %e A298636 Therefore, T(4,2) is less than 3*3*3 = 27: The T(4,2) = 23 drawings are: %e A298636 { [0-1; 0-1; 0-1], [0-1; 0-1; 0-2], [0-1; 0-2; 0-2], [0-2; 0-1; 0-1], %e A298636 [0-2; 0-1; 0-2], [0-2; 0-2; 0-1], [0-2; 0-2; 0-2], [0-1; 0-1; 1-2], %e A298636 [0-2; 0-1; 1-2], [0-2; 0-2; 1-2], [0-1; 1-2; 0-1], [0-1; 1-2; 0-2], %e A298636 [0-2; 1-2; 0-1], [0-2; 1-2; 0-2], [0-1; 1-2; 1-2], [0-2; 1-2; 1-2], %e A298636 [1-2; 0-1; 0-1], [1-2; 0-1; 0-2], [1-2; 0-2; 0-2], [1-2; 0-1; 1-2], %e A298636 [1-2; 1-2; 0-1], [1-2; 1-2; 0-2], [1-2; 1-2; 1-2] } %o A298636 (PARI) A298636(m, n, show=0, c=0)={ my(S, N, u=vector(m-1,i,1)); forvec(a=vector(m-1, i, [0, n-1]), S=Set(a); N=vector(n-1); for(i=1,#a, a[i] && N[a[i]]=if(N[a[i]],concat(N[a[i]],i),i)); forvec(b=vector(m-1, j, [a[j]+1, n]), S=N; for(i=1,#b, b[i]i || b[r]Table of n, a(n) for n = 1..770 (complete sequence). %e A295011 Construct all numbers of the form concat(H,MM,SS) where H < 24 and MM, SS < 60 are primes. These start 2:02:02, 2:02:03, 2:02:03, ... (without ":"s), this is A295014. The corresponding number of seconds after midnight is A292579(HMMSS) = 3600*H + 60*MM + SS. These numbers are listed in A295004. The first prime in that sequence is 7331 = A292579(20211), i.e., the first H:MM:SS for which that number of seconds is prime is 2:02:11, whence a(1) = 20211. %t A295011 With[{s = Prime@ Range@ PrimePi@ 60}, FromDigits@ Flatten[PadLeft[IntegerDigits[#], 2] & /@ #] & /@ Select[Tuples@ {TakeWhile[s, # < 24 &], s, s}, PrimeQ@ NumberCompose[{#1, #2, #3}, {3600, 60, 1}] & @@ # &]] (* _Michael De Vlieger_, Jan 21 2018 *) %o A295011 (PARI) select( t->isprime(A292579(t)), A295014) %Y A295011 Cf. A295014, A295004, A295000, A295002, A295013, A295003; A050246, A159911, A229106; A118848, A118849, A118850. %K A295011 nonn,base,fini,full,new %O A295011 1,1 %A A295011 _M. F. Hasler_, Jan 16 2018 %I A298701 %S A298701 3,11,17,5,13,17,7,13,19,5,17,29,7,13,31,7,23,41,7,19,67,5,29,73,7,31, %T A298701 73,13,37,61,7,11,13,41,13,37,97,7,73,103,3,5,47,89,7,13,19,37,11,13, %U A298701 17,31,7,11,13,101,13,37,241,7,13,19,73,17,41,233,7,13,31,61 %N A298701 Irregular triangle read by rows in which row n lists the prime factors of the n-th Carmichael number. %C A298701 The n-th row is the A002997(n)-th row of A027746. %C A298701 Length of the n-th row is A135717(n). %C A298701 First term of the n-th row is A141710(n). %C A298701 Last term of the n-th row is A081702(n). %H A298701 Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..38296 (rows 1..8032, flattened) %e A298701 Array begins: %e A298701 3, 11, 17, %e A298701 5, 13, 17, %e A298701 7, 13, 19, %e A298701 5, 17, 29, %e A298701 7, 13, 31. %Y A298701 Cf. A002997, A027746, A081702, A135717, A141710. %K A298701 nonn,tabf,new %O A298701 1,1 %A A298701 _Tim Johannes Ohrtmann_, Jan 26 2018 %I A296579 %S A296579 112,240,368,448,496,624,752,880,960,1008,1136,1264,1392,1472,1520, %T A296579 1648,1776,1904,1984,2032,2160,2288,2416,2496,2544,2672,2800,2928, %U A296579 3008,3056,3184,3312,3440,3520,3568,3696,3824,3952,4032,4080,4208,4336,4464,4544,4592 %N A296579 Numbers that are not the sum of 3 squares and a nonnegative 9th power. %C A296579 a(n) consists of the number of forms 16*(8i + 7) (0 <= i <= 152) and 64*(8j + 7) (0 <= j <= 37). %C A296579 The last term in this sequence is a(191) = 19568 = 16*(8*152 + 7) (see A297970). %H A296579 Wikipedia,Legendre's three-square theorem %t A296579 t1=Table[4^2*(8j+7), {j,0,152}]; %t A296579 t2=Table[4^3*(8j+7), {j,0,37}]; %t A296579 t=Union[t1, t2] %Y A296579 Finite subsequence of A004215. %Y A296579 A297970 is subsequence. %Y A296579 Cf. A004771, A022552, A022557, A022561, A022566, A111151. %K A296579 nonn,fini,full,new %O A296579 1,1 %A A296579 _XU Pingya_, Jan 30 2018 %I A298970 %S A298970 1,2,2,4,8,4,8,32,32,8,16,128,219,128,16,32,512,1575,1575,512,32,64, %T A298970 2048,11283,21098,11283,2048,64,128,8192,80972,280468,280468,80972, %U A298970 8192,128,256,32768,581057,3740381,6892031,3740381,581057,32768,256,512,131072 %N A298970 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 5 king-move adjacent elements, with upper left element zero. %C A298970 Table starts %C A298970 ...1.....2.......4.........8...........16.............32...............64 %C A298970 ...2.....8......32.......128..........512...........2048.............8192 %C A298970 ...4....32.....219......1575........11283..........80972...........581057 %C A298970 ...8...128....1575.....21098.......280468........3740381.........49885231 %C A298970 ..16...512...11283....280468......6892031......170137416.......4200575252 %C A298970 ..32..2048...80972...3740381....170137416.....7785598672.....356356552103 %C A298970 ..64..8192..581057..49885231...4200575252...356356552103...30241030285680 %C A298970 .128.32768.4169867.665351771.103715870545.16312003284843.2566506415636896 %H A298970 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A298970 Empirical for column k: %F A298970 k=1: a(n) = 2*a(n-1) %F A298970 k=2: a(n) = 4*a(n-1) %F A298970 k=3: [order 8] %F A298970 k=4: [order 23] %F A298970 k=5: [order 75] %e A298970 Some solutions for n=5 k=4 %e A298970 ..0..0..0..1. .0..0..1..1. .0..0..0..1. .0..0..1..0. .0..0..0..0 %e A298970 ..1..1..0..1. .0..0..0..0. .1..0..0..0. .1..1..1..0. .0..0..1..0 %e A298970 ..0..1..0..1. .0..1..1..1. .1..1..0..1. .1..1..0..1. .0..1..1..0 %e A298970 ..0..0..0..1. .1..1..0..1. .1..1..1..0. .1..0..0..0. .1..1..0..1 %e A298970 ..0..1..1..0. .0..0..1..0. .0..0..0..1. .1..1..1..0. .1..0..0..1 %Y A298970 Column 1 is A000079(n-1). %Y A298970 Column 2 is A004171(n-1). %K A298970 nonn,tabl,new %O A298970 1,2 %A A298970 _R. H. Hardin_, Jan 30 2018 %I A298969 %S A298969 64,8192,581057,49885231,4200575252,356356552103,30241030285680, %T A298969 2566506415636896,217825152899921097,18487503017529807194, %U A298969 1569093086125030344062,133174005805446365581185 %N A298969 Number of nX7 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 5 king-move adjacent elements, with upper left element zero. %C A298969 Column 7 of A298970. %H A298969 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298969 Some solutions for n=5 %e A298969 ..0..0..0..0..1..1..0. .0..0..0..1..1..1..1. .0..0..0..0..0..0..0 %e A298969 ..0..1..1..0..1..1..0. .0..1..1..0..0..0..1. .0..1..1..0..1..1..0 %e A298969 ..0..0..1..0..0..1..1. .0..0..1..0..1..0..0. .0..0..1..0..0..1..0 %e A298969 ..0..0..0..0..1..0..1. .0..0..0..0..1..1..0. .0..0..0..0..1..1..0 %e A298969 ..0..1..1..1..0..0..0. .0..1..1..0..1..1..1. .0..1..1..0..0..1..0 %Y A298969 Cf. A298970. %K A298969 nonn,new %O A298969 1,1 %A A298969 _R. H. Hardin_, Jan 30 2018 %I A298968 %S A298968 32,2048,80972,3740381,170137416,7785598672,356356552103, %T A298968 16312003284843,746698046920333,34181115229827771,1564687555985990215, %U A298968 71625762302968232113,3278769787462125741510,150090290173413011810518 %N A298968 Number of nX6 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 5 king-move adjacent elements, with upper left element zero. %C A298968 Column 6 of A298970. %H A298968 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298968 Some solutions for n=5 %e A298968 ..0..0..1..0..1..0. .0..0..1..1..0..1. .0..0..1..0..1..1. .0..0..1..0..0..0 %e A298968 ..0..0..1..0..1..1. .0..0..1..1..1..1. .0..0..1..1..0..0. .0..0..1..0..1..1 %e A298968 ..0..0..1..1..0..0. .0..0..1..0..0..0. .0..0..1..1..0..1. .0..0..1..0..0..0 %e A298968 ..0..1..0..1..1..0. .0..0..1..0..1..0. .0..1..0..1..1..0. .0..1..0..1..1..0 %e A298968 ..0..0..0..1..0..0. .0..1..0..1..0..1. .0..0..1..0..1..0. .0..1..0..0..0..0 %Y A298968 Cf. A298970. %K A298968 nonn,new %O A298968 1,1 %A A298968 _R. H. Hardin_, Jan 30 2018 %I A298967 %S A298967 16,512,11283,280468,6892031,170137416,4200575252,103715870545, %T A298967 2560893666570,63232497527110,1561310446894473,38551239404752239, %U A298967 951891497499265483,23503717252026765272,580344217887313443874 %N A298967 Number of nX5 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 5 king-move adjacent elements, with upper left element zero. %C A298967 Column 5 of A298970. %H A298967 R. H. Hardin, Table of n, a(n) for n = 1..210 %H A298967 R. H. Hardin, Empirical recurrence of order 75 %F A298967 Empirical recurrence of order 75 (see link above) %e A298967 Some solutions for n=5 %e A298967 ..0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0 %e A298967 ..0..1..0..0..1. .0..0..1..0..1. .0..1..0..1..1. .0..0..1..1..0 %e A298967 ..0..1..1..0..1. .1..1..0..1..0. .1..0..1..1..1. .0..1..1..0..0 %e A298967 ..0..1..0..1..1. .0..0..1..0..0. .1..1..0..1..0. .1..0..1..1..1 %e A298967 ..1..0..0..0..1. .1..0..0..1..1. .0..1..0..1..1. .0..0..1..1..0 %Y A298967 Cf. A298970. %K A298967 nonn,new %O A298967 1,1 %A A298967 _R. H. Hardin_, Jan 30 2018 %I A298966 %S A298966 8,128,1575,21098,280468,3740381,49885231,665351771,8874346832, %T A298966 118364863862,1578735453349,21056975599629,280855308864195, %U A298966 3746013029750793,49963853976933385,666411644915463676 %N A298966 Number of nX4 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 5 king-move adjacent elements, with upper left element zero. %C A298966 Column 4 of A298970. %H A298966 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298966 Empirical: a(n) = 13*a(n-1) +11*a(n-2) -67*a(n-3) -264*a(n-4) -64*a(n-5) +1260*a(n-6) +656*a(n-7) -3045*a(n-8) -3423*a(n-9) +5088*a(n-10) +9482*a(n-11) -514*a(n-12) -10331*a(n-13) -6164*a(n-14) +3337*a(n-15) +4293*a(n-16) +719*a(n-17) -809*a(n-18) +116*a(n-19) +149*a(n-20) -62*a(n-21) -160*a(n-22) -48*a(n-23) %e A298966 Some solutions for n=5 %e A298966 ..0..0..0..1. .0..0..0..1. .0..0..1..0. .0..0..0..1. .0..0..0..1 %e A298966 ..1..0..1..1. .1..0..0..1. .0..1..1..1. .0..1..0..1. .0..1..1..1 %e A298966 ..0..1..0..1. .0..1..1..0. .0..0..1..1. .1..0..1..0. .1..1..1..0 %e A298966 ..0..0..1..1. .1..0..0..1. .0..0..1..0. .1..1..0..0. .1..0..0..0 %e A298966 ..1..1..1..0. .1..0..0..1. .0..1..1..0. .1..1..0..1. .1..0..1..0 %Y A298966 Cf. A298970. %K A298966 nonn,new %O A298966 1,1 %A A298966 _R. H. Hardin_, Jan 30 2018 %I A298965 %S A298965 4,32,219,1575,11283,80972,581057,4169867,29924442,214748731, %T A298965 1541115372,11059607816,79367792570,569572322542,4087459410676, %U A298965 29333104460805,210505091515979,1510661567154766,10841060204501930,77799415112548482 %N A298965 Number of nX3 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 5 king-move adjacent elements, with upper left element zero. %C A298965 Column 3 of A298970. %H A298965 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298965 Empirical: a(n) = 7*a(n-1) +3*a(n-2) -11*a(n-3) -11*a(n-4) +2*a(n-5) +19*a(n-6) -8*a(n-7) -12*a(n-8) %e A298965 Some solutions for n=5 %e A298965 ..0..0..0. .0..1..1. .0..1..1. .0..1..1. .0..0..1. .0..0..1. .0..0..1 %e A298965 ..1..1..1. .1..0..1. .0..0..0. .1..0..1. .0..0..0. .0..1..0. .0..0..0 %e A298965 ..1..1..0. .0..0..1. .0..0..1. .0..0..0. .1..1..1. .0..0..0. .1..1..0 %e A298965 ..0..0..0. .0..0..0. .1..0..1. .0..0..1. .0..1..1. .0..1..1. .0..1..1 %e A298965 ..0..1..0. .1..1..0. .0..0..0. .1..1..1. .0..0..1. .1..1..1. .0..0..1 %Y A298965 Cf. A298970. %K A298965 nonn,new %O A298965 1,1 %A A298965 _R. H. Hardin_, Jan 30 2018 %I A298964 %S A298964 1,8,219,21098,6892031,7785598672,30241030285680,403845370764198967, %T A298964 18541589058526001745929 %N A298964 Number of nXn 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 5 king-move adjacent elements, with upper left element zero. %C A298964 Diagonal of A298970. %e A298964 Some solutions for n=5 %e A298964 ..0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0 %e A298964 ..0..1..1..0..1. .0..0..0..1..0. .0..0..1..1..0. .0..0..1..0..0 %e A298964 ..0..0..1..0..0. .1..1..1..1..0. .1..1..0..1..0. .1..1..0..1..1 %e A298964 ..1..0..0..0..1. .1..0..0..0..1. .1..1..1..1..1. .0..1..0..1..1 %e A298964 ..1..1..1..1..0. .1..1..1..1..1. .0..1..0..0..1. .1..1..1..0..0 %Y A298964 Cf. A298970. %K A298964 nonn,new %O A298964 1,2 %A A298964 _R. H. Hardin_, Jan 30 2018 %I A298963 %S A298963 0,0,0,0,1,0,0,1,1,0,0,2,1,2,0,0,3,2,2,3,0,0,5,3,7,3,5,0,0,8,5,14,14, %T A298963 5,8,0,0,13,8,34,31,34,8,13,0,0,21,13,75,91,91,75,13,21,0,0,34,21,174, %U A298963 230,360,230,174,21,34,0,0,55,34,396,633,1144,1144,633,396,34,55,0,0,89,55 %N A298963 T(n,k)=Number of nXk 0..1 arrays with every element equal to 3, 5, 6, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298963 Table starts %C A298963 .0..0..0...0....0.....0.....0......0.......0........0.........0.........0 %C A298963 .0..1..1...2....3.....5.....8.....13......21.......34........55........89 %C A298963 .0..1..1...2....3.....5.....8.....13......21.......34........55........89 %C A298963 .0..2..2...7...14....34....75....174.....396......907......2070......4734 %C A298963 .0..3..3..14...31....91...230....633....1685.....4552.....12185.....32765 %C A298963 .0..5..5..34...91...360..1144...4062...13794....47972....164529....567553 %C A298963 .0..8..8..75..230..1144..4263..18517...75262...317141...1307739...5446889 %C A298963 .0.13.13.174..633..4062.18517.101720..514070..2720154..14017132..73173545 %C A298963 .0.21.21.396.1685.13794.75262.514070.3172536.20741504.130945779.840773447 %H A298963 R. H. Hardin, Table of n, a(n) for n = 1..312 %F A298963 Empirical for column k: %F A298963 k=1: a(n) = a(n-1) %F A298963 k=2: a(n) = a(n-1) +a(n-2) %F A298963 k=3: a(n) = a(n-1) +a(n-2) %F A298963 k=4: a(n) = 2*a(n-1) +a(n-3) +2*a(n-4) -2*a(n-5) %F A298963 k=5: [order 11] %F A298963 k=6: [order 30] %e A298963 Some solutions for n=5 k=4 %e A298963 ..0..0..1..1. .0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..1..1 %e A298963 ..0..0..1..1. .0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..1..1 %e A298963 ..0..0..0..0. .0..0..0..0. .1..1..1..1. .0..0..0..0. .0..0..0..0 %e A298963 ..1..1..0..0. .1..1..0..0. .1..1..0..0. .1..1..1..1. .0..0..0..0 %e A298963 ..1..1..0..0. .1..1..0..0. .1..1..0..0. .1..1..1..1. .0..0..0..0 %Y A298963 Columns 2 and 3 are A000045(n-1). %K A298963 nonn,tabl,new %O A298963 1,12 %A A298963 _R. H. Hardin_, Jan 30 2018 %I A298962 %S A298962 0,8,8,75,230,1144,4263,18517,75262,317141,1307739,5446889,22584896, %T A298962 93917334,389965508,1620245230,6729536166,27956779018,116131464818, %U A298962 482427395426,2004019363266,8324893628475,34582253578090,143657911533546 %N A298962 Number of nX7 0..1 arrays with every element equal to 3, 5, 6, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298962 Column 7 of A298963. %H A298962 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298962 Some solutions for n=5 %e A298962 ..0..0..0..0..0..1..1. .0..0..1..1..0..0..0. .0..0..1..1..1..0..0 %e A298962 ..0..0..0..0..0..1..1. .0..0..1..1..0..0..0. .0..0..1..1..1..0..0 %e A298962 ..0..0..0..0..0..0..0. .1..1..1..1..0..0..0. .0..0..1..1..1..0..0 %e A298962 ..1..1..1..1..0..0..0. .1..1..1..1..1..1..1. .0..0..1..1..1..1..1 %e A298962 ..1..1..1..1..0..0..0. .1..1..1..1..1..1..1. .0..0..1..1..1..1..1 %Y A298962 Cf. A298963. %K A298962 nonn,new %O A298962 1,2 %A A298962 _R. H. Hardin_, Jan 30 2018 %I A298961 %S A298961 0,5,5,34,91,360,1144,4062,13794,47972,164529,567553,1953217,6732815, %T A298961 23189914,79894441,275211747,948122291,3266225384,11252104514, %U A298961 38762888085,133536840437,460029356546,1584785398981,5459526766694 %N A298961 Number of nX6 0..1 arrays with every element equal to 3, 5, 6, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298961 Column 6 of A298963. %H A298961 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298961 Empirical: a(n) = 4*a(n-1) -a(n-2) -3*a(n-3) +14*a(n-4) -44*a(n-5) -22*a(n-6) -27*a(n-7) +33*a(n-8) +210*a(n-9) +164*a(n-10) +294*a(n-11) -198*a(n-12) -391*a(n-13) -686*a(n-14) -638*a(n-15) +36*a(n-16) +450*a(n-17) +656*a(n-18) +232*a(n-19) +286*a(n-20) -8*a(n-21) +96*a(n-22) -142*a(n-23) +20*a(n-24) -196*a(n-25) -32*a(n-26) +16*a(n-27) +24*a(n-28) -32*a(n-29) +16*a(n-30) %e A298961 Some solutions for n=5 %e A298961 ..0..0..0..1..1..1. .0..0..0..1..1..1. .0..0..1..1..0..0. .0..0..0..0..0..0 %e A298961 ..0..0..0..1..1..1. .0..0..0..1..1..1. .0..0..1..1..0..0. .0..0..0..0..0..0 %e A298961 ..1..1..1..1..1..1. .0..0..0..0..0..0. .0..0..1..1..1..1. .0..0..0..0..0..0 %e A298961 ..1..1..1..1..1..1. .1..1..1..0..0..0. .0..0..1..1..1..1. .0..0..0..0..1..1 %e A298961 ..1..1..1..1..1..1. .1..1..1..0..0..0. .0..0..1..1..1..1. .0..0..0..0..1..1 %Y A298961 Cf. A298963. %K A298961 nonn,new %O A298961 1,2 %A A298961 _R. H. Hardin_, Jan 30 2018 %I A298960 %S A298960 0,3,3,14,31,91,230,633,1685,4552,12185,32765,87980,236467,635199, %T A298960 1706614,4584923,12318567,33096042,88919069,238897433,641845172, %U A298960 1724442741,4633054817,12447607328,33442938823,89851018539,241402408354 %N A298960 Number of nX5 0..1 arrays with every element equal to 3, 5, 6, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298960 Column 5 of A298963. %H A298960 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298960 Empirical: a(n) = 3*a(n-1) -a(n-2) +5*a(n-4) -9*a(n-5) -a(n-6) -8*a(n-7) +8*a(n-9) -6*a(n-10) +2*a(n-11) %e A298960 Some solutions for n=5 %e A298960 ..0..0..1..1..1. .0..0..1..1..1. .0..0..0..1..1. .0..0..1..1..1 %e A298960 ..0..0..1..1..1. .0..0..1..1..1. .0..0..0..1..1. .0..0..1..1..1 %e A298960 ..0..0..0..0..0. .1..1..1..1..1. .1..1..1..1..1. .0..0..1..1..1 %e A298960 ..0..0..0..0..0. .1..1..1..0..0. .1..1..1..0..0. .0..0..0..0..0 %e A298960 ..0..0..0..0..0. .1..1..1..0..0. .1..1..1..0..0. .0..0..0..0..0 %Y A298960 Cf. A298963. %K A298960 nonn,new %O A298960 1,2 %A A298960 _R. H. Hardin_, Jan 30 2018 %I A298959 %S A298959 0,2,2,7,14,34,75,174,396,907,2070,4734,10819,24730,56520,129187, %T A298959 295274,674890,1542547,3525702,8058468,18418715,42098446,96221670, %U A298959 219927587,502674114,1148929360,2626032755,6002151458,13718725330,31355993907 %N A298959 Number of nX4 0..1 arrays with every element equal to 3, 5, 6, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298959 Column 4 of A298963. %H A298959 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298959 Empirical: a(n) = 2*a(n-1) +a(n-3) +2*a(n-4) -2*a(n-5) %e A298959 Some solutions for n=5 %e A298959 ..0..0..0..0. .0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..1..1 %e A298959 ..0..0..0..0. .0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..1..1 %e A298959 ..1..1..1..1. .0..0..0..0. .0..0..1..1. .1..1..0..0. .0..0..1..1 %e A298959 ..1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..0..0. .0..0..1..1 %e A298959 ..1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..0..0. .0..0..1..1 %Y A298959 Cf. A298963. %K A298959 nonn,new %O A298959 1,2 %A A298959 _R. H. Hardin_, Jan 30 2018 %I A298958 %S A298958 0,1,1,7,31,360,4263,101720,3172536,169440015,12802881385, %T A298958 1552491730921 %N A298958 Number of nXn 0..1 arrays with every element equal to 3, 5, 6, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298958 Diagonal of A298963. %e A298958 Some solutions for n=5 %e A298958 ..0..0..1..1..1. .0..0..0..1..1. .0..0..0..0..0. .0..0..0..1..1 %e A298958 ..0..0..1..1..1. .0..0..0..1..1. .0..0..0..0..0. .0..0..0..1..1 %e A298958 ..1..1..1..1..1. .0..0..0..0..0. .0..0..0..0..0. .0..0..0..1..1 %e A298958 ..1..1..1..0..0. .1..1..0..0..0. .1..1..1..0..0. .1..1..0..0..0 %e A298958 ..1..1..1..0..0. .1..1..0..0..0. .1..1..1..0..0. .1..1..0..0..0 %Y A298958 Cf. A298963. %K A298958 nonn,new %O A298958 1,4 %A A298958 _R. H. Hardin_, Jan 30 2018 %I A298957 %S A298957 0,0,0,0,1,0,0,1,1,0,0,2,1,2,0,0,3,2,2,3,0,0,5,3,4,3,5,0,0,8,5,6,6,5, %T A298957 8,0,0,13,8,11,10,11,8,13,0,0,21,13,18,22,22,18,13,21,0,0,34,21,31,47, %U A298957 102,47,31,21,34,0,0,55,34,53,143,390,390,143,53,34,55,0,0,89,55,91,418,1990 %N A298957 T(n,k)=Number of nXk 0..1 arrays with every element equal to 3, 4, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298957 Table starts %C A298957 .0..0..0..0...0....0......0.......0........0..........0...........0 %C A298957 .0..1..1..2...3....5......8......13.......21.........34..........55 %C A298957 .0..1..1..2...3....5......8......13.......21.........34..........55 %C A298957 .0..2..2..4...6...11.....18......31.......53.........91.........156 %C A298957 .0..3..3..6..10...22.....47.....143......418.......1449........4821 %C A298957 .0..5..5.11..22..102....390....1990.....8875......42338......200422 %C A298957 .0..8..8.18..47..390...3101...21928...156972....1153968.....8497555 %C A298957 .0.13.13.31.143.1990..21928..300849..3354122...43052414...516822488 %C A298957 .0.21.21.53.418.8875.156972.3354122.54865647.1090267927.19403449274 %H A298957 R. H. Hardin, Table of n, a(n) for n = 1..220 %F A298957 Empirical for column k: %F A298957 k=1: a(n) = a(n-1) %F A298957 k=2: a(n) = a(n-1) +a(n-2) %F A298957 k=3: a(n) = a(n-1) +a(n-2) %F A298957 k=4: a(n) = 2*a(n-1) +a(n-2) -a(n-3) -2*a(n-4) -2*a(n-5) +a(n-6) +a(n-7) %F A298957 k=5: [order 69] %e A298957 Some solutions for n=5 k=4 %e A298957 ..0..0..0..0. .0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..1..1 %e A298957 ..0..0..0..0. .0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..1..1 %e A298957 ..0..0..0..0. .0..0..0..0. .0..0..1..1. .1..1..1..1. .1..1..0..0 %e A298957 ..1..1..1..1. .0..0..0..0. .1..1..0..0. .1..1..1..1. .1..1..0..0 %e A298957 ..1..1..1..1. .0..0..0..0. .1..1..0..0. .1..1..1..1. .1..1..0..0 %Y A298957 Columns 2 and 3 are A000045(n-1). %Y A298957 Column 4 is A298163. %K A298957 nonn,tabl,new %O A298957 1,12 %A A298957 _R. H. Hardin_, Jan 30 2018 %I A298956 %S A298956 0,8,8,18,47,390,3101,21928,156972,1153968,8497555,64840157,491050746, %T A298956 3752672242,28653256675,219001866361,1673330811892,12795534191502, %U A298956 97817273521916,748006580822607,5719773571757293,43742434623477117 %N A298956 Number of nX7 0..1 arrays with every element equal to 3, 4, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298956 Column 7 of A298957. %H A298956 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298956 Some solutions for n=5 %e A298956 ..0..0..0..1..1..1..1. .0..0..1..1..1..1..1. .0..0..0..0..0..0..0 %e A298956 ..0..0..1..0..1..1..1. .0..0..1..1..0..1..1. .0..0..0..0..0..0..0 %e A298956 ..1..1..0..1..0..0..0. .1..1..0..0..0..0..1. .0..0..0..0..0..0..0 %e A298956 ..1..1..1..0..0..0..0. .1..1..1..0..0..1..1. .1..1..1..1..1..1..1 %e A298956 ..1..1..1..0..0..0..0. .1..1..1..1..1..1..1. .1..1..1..1..1..1..1 %Y A298956 Cf. A298957. %K A298956 nonn,new %O A298956 1,2 %A A298956 _R. H. Hardin_, Jan 30 2018 %I A298955 %S A298955 0,5,5,11,22,102,390,1990,8875,42338,200422,1001873,4864576,24344988, %T A298955 120791381,599718216,2979586795,14824136258,73594777737,365973623961, %U A298955 1819455689317,9043962959653,44964812729586,223577501339948 %N A298955 Number of nX6 0..1 arrays with every element equal to 3, 4, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298955 Column 6 of A298957. %H A298955 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298955 Some solutions for n=5 %e A298955 ..0..0..0..1..1..1. .0..0..1..1..1..1. .0..0..1..1..1..1. .0..0..0..1..1..1 %e A298955 ..0..0..0..1..1..1. .0..0..1..1..1..1. .0..0..1..1..1..1. .0..0..0..1..1..1 %e A298955 ..0..0..0..1..1..1. .1..1..0..0..0..0. .0..0..1..1..1..1. .0..0..1..0..1..1 %e A298955 ..1..1..1..0..0..0. .1..1..0..0..0..0. .1..1..0..0..0..0. .1..1..0..1..0..0 %e A298955 ..1..1..1..0..0..0. .1..1..0..0..0..0. .1..1..0..0..0..0. .1..1..1..0..0..0 %Y A298955 Cf. A298957. %K A298955 nonn,new %O A298955 1,2 %A A298955 _R. H. Hardin_, Jan 30 2018 %I A298954 %S A298954 0,3,3,6,10,22,47,143,418,1449,4821,17127,59477,211515,744552,2643659, %T A298954 9347408,33151822,117393355,416155667,1474364490,5225593450, %U A298954 18516679195,65623399116,232549566847,824133935862,2920552934068,10350044812936 %N A298954 Number of nX5 0..1 arrays with every element equal to 3, 4, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298954 Column 5 of A298957. %H A298954 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298954 Empirical: a(n) = 3*a(n-1) +11*a(n-2) -21*a(n-3) -64*a(n-4) +8*a(n-5) +306*a(n-6) +158*a(n-7) -966*a(n-8) -253*a(n-9) +1866*a(n-10) +538*a(n-11) -3171*a(n-12) -2696*a(n-13) +5783*a(n-14) +6836*a(n-15) -8539*a(n-16) -10968*a(n-17) +3344*a(n-18) +18183*a(n-19) +17794*a(n-20) -40229*a(n-21) -42832*a(n-22) +77333*a(n-23) +58969*a(n-24) -100771*a(n-25) -71266*a(n-26) +98796*a(n-27) +81254*a(n-28) -84064*a(n-29) -77712*a(n-30) +44321*a(n-31) +23231*a(n-32) +53723*a(n-33) -8064*a(n-34) -69827*a(n-35) +17009*a(n-36) -52317*a(n-37) -1691*a(n-38) +142405*a(n-39) +48528*a(n-40) +77582*a(n-41) -125126*a(n-42) -341214*a(n-43) +132239*a(n-44) +122239*a(n-45) +76573*a(n-46) +54854*a(n-47) -276162*a(n-48) -73361*a(n-49) +176348*a(n-50) +121504*a(n-51) -15871*a(n-52) -53527*a(n-53) -14754*a(n-54) +30249*a(n-55) +27322*a(n-56) -10822*a(n-57) -22534*a(n-58) -5620*a(n-59) +3200*a(n-60) -1282*a(n-61) -3509*a(n-62) -2630*a(n-63) -946*a(n-64) +30*a(n-65) +121*a(n-66) +70*a(n-67) +22*a(n-68) +2*a(n-69) %e A298954 Some solutions for n=5 %e A298954 ..0..0..1..1..1. .0..0..0..1..1. .0..0..1..1..1. .0..0..0..1..1 %e A298954 ..0..0..1..1..1. .0..0..0..1..1. .0..0..1..1..1. .0..0..0..1..1 %e A298954 ..1..1..0..0..0. .1..1..1..0..0. .0..0..1..1..1. .0..0..0..1..1 %e A298954 ..1..1..0..0..0. .1..1..1..0..0. .0..0..1..1..1. .1..1..1..0..0 %e A298954 ..1..1..0..0..0. .1..1..1..0..0. .0..0..1..1..1. .1..1..1..0..0 %Y A298954 Cf. A298957. %K A298954 nonn,new %O A298954 1,2 %A A298954 _R. H. Hardin_, Jan 30 2018 %I A298953 %S A298953 0,1,1,4,10,102,3101,300849,54865647,36049757678 %N A298953 Number of nXn 0..1 arrays with every element equal to 3, 4, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298953 Diagonal of A298957. %e A298953 Some solutions for n=5 %e A298953 ..0..0..0..0..0. .0..0..1..1..1. .0..0..0..1..1. .0..0..0..0..0 %e A298953 ..0..0..0..0..0. .0..0..1..1..1. .0..0..0..1..1. .0..0..0..0..0 %e A298953 ..0..0..0..0..0. .0..0..1..1..1. .0..0..0..1..1. .0..0..0..0..0 %e A298953 ..1..1..1..1..1. .0..0..1..1..1. .0..0..0..1..1. .0..0..0..0..0 %e A298953 ..1..1..1..1..1. .0..0..1..1..1. .0..0..0..1..1. .0..0..0..0..0 %Y A298953 Cf. A298957. %K A298953 nonn,new %O A298953 1,4 %A A298953 _R. H. Hardin_, Jan 30 2018 %I A298804 %S A298804 0,1,1,3,2,1,9,6,4,3,31,22,16,12,9,121,90,68,52,40,31,523,402,312,244, %T A298804 192,152,121 %N A298804 Triangle T(n,k) (1 <= k <= n) read by rows: A046936 with rows reversed and offset changed to 1. %C A298804 This is another version of Moser's version (A046936) of Aitken's array (A011971). %C A298804 Although offset 0 is better for A011971 and A046936, for this version offset 1 is more appropriate. %C A298804 Comments from _Don Knuth_, Jan 29 2018 (Start): %C A298804 a(n,k) is the number of set partitions (i.e. equivalence classes) in which (i) 1 is not equivalent to 2, ..., nor k; and (ii) the last part, when parts are ordered by their smallest element, has size 1; (iii) that last part isn't simply "1". (Equivalently, n>1.) %C A298804 It's not difficult to prove this characterization of a(k,n). For example, if we know that there are 22 partitions of {1,2,3,4,5} with 1 inequivalent to 2, and 6 partitions of {1,2,3,4} with %C A298804 1 inequivalent to 2, then there are 6 partitions of {1,2,3,4,5} with 1 inequivalent to 2 and 1 equivalent to 3. Hence there are 16 with 1 equivalent to neither 2 nor 3. %C A298804 The same property, but leaving out conditions (ii) and (iii), characterizes Pierce's triangular array A123346. (End) %H A298804 Don Knuth, Email to N. J. A. Sloane, Jan 29 2018 %e A298804 Triangle begins: %e A298804 0, %e A298804 1, 1, %e A298804 3, 2, 1, %e A298804 9, 6, 4, 3, %e A298804 31, 22, 16, 12, 9, %e A298804 121, 90, 68, 52, 40, 31 %e A298804 523, 402, 312, 244, 192, 152, 121 %e A298804 ... %Y A298804 Cf. A011971, A040027, A046936, A123346. %K A298804 nonn,tabl,more,new %O A298804 1,4 %A A298804 _N. J. A. Sloane_, Jan 30 2018, following a suggestion from _Don Knuth_, Jan 29 2018. %I A297776 %S A297776 1,1,1,1,1,1,1,2,1,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2, %T A297776 2,1,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,1,2,2,3,3,3, %U A297776 3,3,3,2,1,2,2,2,2,2,2,3,2,2,3,3,3,3 %N A297776 Number of distinct runs in base-8 digits of n. %C A297776 Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. %H A297776 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297776 1262656 in base-8: 1,0,0,1,0,0,1; five runs, of which 2 are distinct, so that a(1262656) = 2. %t A297776 b = 8; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297776 Table[s[n], {n, 1, 200}] %Y A297776 Cf. A043560 (number of runs, not necessarily distinct), A297770, A043535. %K A297776 nonn,base,easy,new %O A297776 1,8 %A A297776 _Clark Kimberling_, Jan 29 2018 %I A298939 %S A298939 1,1,1,4,1,286,7582,202028,6473625,226029577,8338249868,391526193477, %T A298939 19990594900630,1159906506684446,74890158861242740, %U A298939 5119732406649036418,380146984328280974281,30198665638519565614034,2555354508318427693497565 %N A298939 Number of ordered ways of writing n^3 as a sum of n squares of positive integers. %H A298939 Index entries for sequences related to sums of squares %F A298939 a(n) = [x^(n^3)] (Sum_{k>=1} x^(k^2))^n. %e A298939 a(3) = 4 because we have [25, 1, 1], [9, 9, 9], [1, 25, 1] and [1, 1, 25]. %t A298939 Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^3}], {n, 0, 18}] %Y A298939 Cf. A000290, A000578, A006456, A030273, A037444, A066535, A218494, A232173, A287617, A298329, A298330, A298935, A298937, A298938. %K A298939 nonn,new %O A298939 0,4 %A A298939 _Ilya Gutkovskiy_, Jan 29 2018 %I A298938 %S A298938 1,1,1,4,5,686,13942,455988,13617853,454222894,18323165948, %T A298938 802161109047,42149084452070,2481730049781672,157265294178424356, %U A298938 10977302934685469078,812821237985857557677,64539935903231450294134,5504599828399250884049308 %N A298938 Number of ordered ways of writing n^3 as a sum of n squares of nonnegative integers. %H A298938 Index entries for sequences related to sums of squares %F A298938 a(n) = [x^(n^3)] (Sum_{k>=0} x^(k^2))^n. %e A298938 a(4) = 5 because we have [64, 0, 0, 0], [16, 16, 16, 16], [0, 64, 0, 0], [0, 0, 64, 0] and [0, 0, 0, 64]. %t A298938 Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^3}], {n, 0, 18}] %Y A298938 Cf. A000290, A000578, A006456, A030273, A037444, A066535, A218494, A232173, A287617, A298329, A298330, A298935, A298936, A298939. %K A298938 nonn,new %O A298938 0,4 %A A298938 _Ilya Gutkovskiy_, Jan 29 2018 %I A298937 %S A298937 1,1,0,0,0,0,0,7,1,0,0,9240,34650,1716,48477,551915,6726720,89973520, %T A298937 102639744,1824625081,9915389400,30143458884,278196062760, %U A298937 1995766236541,6611689457736,64547920386450,236756174748626,2315743488707806 %N A298937 Number of ordered ways of writing n^2 as a sum of n positive cubes. %H A298937 Index entries for sequences related to sums of cubes %F A298937 a(n) = [x^(n^2)] (Sum_{k>=1} x^(k^3))^n. %e A298937 a(7) = 7 because we have [8, 8, 8, 8, 8, 8, 1], [8, 8, 8, 8, 8, 1, 8], [8, 8, 8, 8, 1, 8, 8], [8, 8, 8, 1, 8, 8, 8], [8, 8, 1, 8, 8, 8, 8], [8, 1, 8, 8, 8, 8, 8] and [1, 8, 8, 8, 8, 8, 8]. %t A298937 Join[{1}, Table[SeriesCoefficient[Sum[x^k^3, {k, 1, Floor[n^(2/3) + 1]}]^n, {x, 0, n^2}], {n, 1, 27}]] %Y A298937 Cf. A000290, A000578, A023358, A030272, A218495, A259792, A291700, A298671, A298672, A298934, A298936. %K A298937 nonn,new %O A298937 0,8 %A A298937 _Ilya Gutkovskiy_, Jan 29 2018 %I A298936 %S A298936 1,1,0,6,6,20,120,7,1689,6636,36540,64020,963996,2894892,19555965, %T A298936 176079995,955611188,6684303780,42462792168,292378003753, %U A298936 1886275214112,13384059605364,87399249887334,624073002367892,5080120229014734,37587589611771480 %N A298936 Number of ordered ways of writing n^2 as a sum of n nonnegative cubes. %H A298936 Index entries for sequences related to sums of cubes %F A298936 a(n) = [x^(n^2)] (Sum_{k>=0} x^(k^3))^n. %e A298936 a(3) = 6 because we have [8, 1, 0], [8, 0, 1], [1, 8, 0], [1, 0, 8], [0, 8, 1] and [0, 1, 8]. %p A298936 f:= n -> coeff(add(x^(k^3),k=0..floor(n^(2/3)))^n,x,n^2): %p A298936 map(f, [$0..30]); # _Robert Israel_, Jan 29 2018 %t A298936 Table[SeriesCoefficient[Sum[x^k^3, {k, 0, Floor[n^(2/3) + 1]}]^n, {x, 0, n^2}], {n, 0, 25}] %Y A298936 Cf. A000290, A000578, A023358, A030272, A218495, A259792, A291700, A298671, A298672, A298934, A298937. %K A298936 nonn,new %O A298936 0,4 %A A298936 _Ilya Gutkovskiy_, Jan 29 2018 %I A298935 %S A298935 1,1,0,0,1,5,8,40,96,297,1269,3456,12839,46691,153111,577167,2054576, %T A298935 7602937,29000337,110645967,418889453,1580667760,6058528796, %U A298935 23121913246,89793473393,350029321425,1359919742613,5340642744919,20948242218543,82505892314268 %N A298935 Number of partitions of n^3 into distinct squares. %H A298935 Alois P. Heinz, Table of n, a(n) for n = 0..80 %H A298935 Index entries for sequences related to sums of squares %H A298935 Index entries for related partition-counting sequences %F A298935 a(n) = [x^(n^3)] Product_{k>=1} (1 + x^(k^2)). %F A298935 a(n) = A033461(A000578(n)). %e A298935 a(5) = 5 because we have [121, 4], [100, 25], [100, 16, 9], [64, 36, 25] and [64, 36, 16, 9]. %t A298935 Table[SeriesCoefficient[Product[1 + x^k^2, {k, 1, Floor[n^(3/2) + 1]}], {x, 0, n^3}], {n, 0, 29}] %Y A298935 Cf. A000290, A000578, A030272, A030273, A033461, A037444, A218494, A298330, A298642, A298934. %K A298935 nonn,new %O A298935 0,6 %A A298935 _Ilya Gutkovskiy_, Jan 29 2018 %I A298673 %S A298673 1,1,1,4,3,1,26,19,6,1,236,170,55,10,1,2752,1966,645,125,15,1,39208, %T A298673 27860,9226,1855,245,21,1,660032,467244,155764,32081,4480,434,28,1, %U A298673 12818912,9049584,3031876,635124,92001,9576,714,36,1,282137824,198754016,66845340,14180440,2108085,230097,18690,1110,45,1 %N A298673 Inverse matrix of A135494. %C A298673 Since this is the inverse matrix of A135494 with row polynomials q_n(t), first introduced in that entry by _R. J. Mathar_, and the row polynomials p_n(t) of this entry are a binomial Sheffer polynomial sequence, the row polynomials of the inverse pair are umbral compositional inverses, i.e., p_n(q.(t)) = q_n(p.(t)) = t^n. For example, p_3(q.(t)) = 4q_1(t) + 3q_2(t) + q_3(t) = 4t + 3(-t + t^2) + (-t -3t^2 +t^3) = t^3. In addition, both sequences possess the umbral convolution property (p.x) + p.(y))^n = p_n(x+y) with p_0(t) = 1. %C A298673 This is the inverse of the Bell matrix generated by A153881; for the definition of the Bell matrix see the link. - _Peter Luschny_, Jan 26 2018 %H A298673 Peter Luschny, The Bell transform %F A298673 E.g.f.: e^[p.(t)x] = e^[t*h(x)] = exp[t*[(x-1)/2 + T{ (1/2) * exp[(x-1)/2] }], where T is the tree function of A000169 related to the Lambert function. h(x) = sum(j=1,...) A000311(j) * x^j / j! = exp[xp.'(0)], so the first column of this entry's matrix is A000311(n) for n > 0 and the second column of the full matrix for p_n(t) to n >= 0. The compositional inverse of h(x) is h^(-1)(x) = 1 + 2x - e^x. %F A298673 The lowering operator is L = h^(-1)(D) = 1 + 2D - e^D with D = d/dt, i.e., L p_n(t) = n * p_(n-1)(t). For example, L p_3(t) = (D - D^2! - D^3/3! - ...) (4t + 6t^ + t^3) = 3 (t + t^2) = 3 p_2(t). %F A298673 The raising operator is R = t * 1/[d[h^(-1)(D)]/dD] = t * 1/[2 - e^D)] = t (1 + D + 3D^2/2! + 13D^3/3! + ...). The coefficients of R are A000670. For example, R p_2(t) = t (1 + D + 3D^2/2! + ...) (t + t^2) = 4t + 3t^2 + t^3 = p_3(t). %F A298673 The row sums are A006351, or essentially 2*A000311. %e A298673 Matrix begins as %e A298673 1; %e A298673 1; 1; %e A298673 4, 3, 1; %e A298673 26, 19, 6, 1; %e A298673 236, 170, 55, 10, 1; %e A298673 2752, 1966, 645, 125, 15, 1; %p A298673 # The function BellMatrix is defined in A264428. Adds (1,0,0,0, ..) as column 0. %p A298673 BellMatrix(n -> `if`(n=0, 1, -1), 9): MatrixInverse(%); # _Peter Luschny_, Jan 26 2018 %Y A298673 Cf. A000311, A000169, A000670, A006351, A135494. %K A298673 nonn,tabl,new %O A298673 1,4 %A A298673 _Tom Copeland_, Jan 24 2018 %I A298934 %S A298934 1,1,0,1,0,0,1,0,1,0,1,0,0,0,0,2,0,1,0,0,0,1,0,0,1,0,0,3,1,0,0,0,0,0, %T A298934 0,3,3,1,0,3,0,2,4,0,0,1,0,0,2,3,1,1,0,6,3,6,1,6,0,3,9,0,6,6,7,0,10,3, %U A298934 3,6,0,8,6,13,2,10,9,10,19,2,14,21,7,2,25 %N A298934 Number of partitions of n^2 into distinct cubes. %H A298934 Alois P. Heinz, Table of n, a(n) for n = 0..1000 %H A298934 Index entries for sequences related to sums of cubes %H A298934 Index entries for related partition-counting sequences %F A298934 a(n) = [x^(n^2)] Product_{k>=1} (1 + x^(k^3)). %F A298934 a(n) = A279329(A000290(n)). %e A298934 a(15) = 2 because we have [216, 8, 1] and [125, 64, 27, 8, 1]. %p A298934 b:= proc(n, i) option remember; `if`(n=0, 1, %p A298934 `if`(n>i^2*(i+1)^2/4, 0, b(n, i-1)+ %p A298934 `if`(i^3>n, 0, b(n-i^3, i-1)))) %p A298934 end: %p A298934 a:= n-> b(n^2, n): %p A298934 seq(a(n), n=0..100); # _Alois P. Heinz_, Jan 29 2018 %t A298934 Table[SeriesCoefficient[Product[1 + x^k^3, {k, 1, Floor[n^(2/3) + 1]}], {x, 0, n^2}], {n, 0, 84}] %Y A298934 Cf. A000290, A000578, A030272, A030273, A218495, A259792, A279329, A298672, A298848, A298935. %K A298934 nonn,new %O A298934 0,16 %A A298934 _Ilya Gutkovskiy_, Jan 29 2018 %I A298933 %S A298933 1,2,3,4,4,6,5,6,6,4,8,6,9,6,6,12,8,12,8,8,9,8,12,6,8,14,12,12,8,12, %T A298933 13,12,18,8,8,12,16,14,12,12,16,12,13,14,6,20,16,18,8,10,18,16,20,12, %U A298933 16,16,15,20,12,18,24,14,18,8,16,18,16,22,12,12,20,24 %N A298933 Expansion of f(x, x^2) * f(x, x^3) * f(x^2, x^4) in powers of x where f(, ) is Ramanujan's general theta function. %C A298933 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). %H A298933 Robert Israel, Table of n, a(n) for n = 0..10000 %H A298933 M. Somos, Introduction to Ramanujan theta functions %H A298933 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions %F A298933 Expansion of phi(x) * phi(-x^3) * phi(-x^6) / chi(-x^2)^3 in powers of x where phi(), chi() are Ramanujan theta functions. %F A298933 Expansion of q^(-1/4) * eta(q^2)^2 * eta(q^3)^2 * eta(q^4) * eta(q^6) / (eta(q)^2 * eta(q^12)) in powers of q. %F A298933 Euler transform of period 12 sequence [2, 0, 0, -1, 2, -3, 2, -1, 0, 0, 2, -3, ...]. %F A298933 a(n) = A298932(2*n). %e A298933 G.f. = 1 + 2*x + 3*x^2 + 4*x^3 + 4*x^4 + 6*x^5 + 5*x^6 + 6*x^7 + 6*x^8 + ... %e A298933 G.f. = q + 2*q^5 + 3*q^9 + 4*q^13 + 4*q^17 + 6*q^21 + 5*q^25 + 6*q^29 + ... %p A298933 N:= 100: %p A298933 S:= series(JacobiTheta3(0,x)*JacobiTheta4(0,x^3)*JacobiTheta4(0,x^6)*expand(QDifferenceEquations:-QPochhammer(-x^2,x^2,floor(N/2)))^3, x, N+1): %p A298933 seq(coeff(S,x,j),j=0..N); # _Robert Israel_, Jan 29 2018 %t A298933 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^6] QPochhammer[ -x^2, x^2]^3, {x, 0, n}]; %o A298933 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^6 + A) / (eta(x + A)^2 * eta(x^12 + A)), n))}; %Y A298933 Cf. A298932. %K A298933 nonn,new %O A298933 0,2 %A A298933 _Michael Somos_, Jan 29 2018 %I A298932 %S A298932 1,1,2,0,3,2,4,0,4,4,6,0,5,3,6,0,6,4,4,0,8,4,6,0,9,6,6,0,6,6,12,0,8,4, %T A298932 12,0,8,7,8,0,9,6,8,0,12,8,6,0,8,6,14,0,12,6,12,0,8,8,12,0,13,6,12,0, %U A298932 18,10,8,0,8,12,12,0,16,7,14,0,12,8,12,0,16 %N A298932 Expansion of f(-x^3)^3 * phi(-x^12) / (f(-x) * chi(-x^4)) in powers of x where phi(), chi(), f() are Ramanujan theta functions. %C A298932 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). %H A298932 M. Somos, Introduction to Ramanujan theta functions %H A298932 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions %F A298932 Expansion of q^(-1/2) * eta(q^3)^3 * eta(q^8) * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^24)) in powers of q. %F A298932 Euler transform of period 24 sequence [1, 1, -2, 2, 1, -2, 1, 1, -2, 1, 1, -3, 1, 1, -2, 1, 1, -2, 1, 2, -2, 1, 1, -3, ...]. %F A298932 a(4*n + 3) = 0. a(3*n + 2) = 2 * A213607(n). a(n) = A298931(3*n). a(2*n) = A298933(n). %e A298932 G.f. = 1 + x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^6 + 4*x^8 + 4*x^9 + ... %e A298932 G.f. = q + q^3 + 2*q^5 + 3*q^9 + 2*q^11 + 4*q^13 + 4*q^17 + 4*q^19 + ... %t A298932 a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 QPochhammer[ -x^4, x^4] EllipticTheta[ 4, 0, x^12] / QPochhammer[ x], {x, 0, n}]; %o A298932 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^24 + A)), n))}; %Y A298932 Cf. A213607, A298931, A298933. %K A298932 nonn,new %O A298932 0,3 %A A298932 _Michael Somos_, Jan 29 2018 %I A298931 %S A298931 1,0,0,1,1,0,2,1,0,0,2,0,3,0,0,2,2,0,4,1,0,0,2,0,4,0,0,4,1,0,6,2,0,0, %T A298931 2,0,5,0,0,3,3,0,6,1,0,0,4,0,6,0,0,4,5,0,4,3,0,0,2,0,8,0,0,4,3,0,6,3, %U A298931 0,0,4,0,9,0,0,6,4,0,6,2,0,0,4,0,6,0,0 %N A298931 Expansion of psi(x^4) * c(x^3) / (3*x) where phi() is a Ramanujan theta function and c() is a cubic AGM theta function. %C A298931 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). %C A298931 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882). %H A298931 M. Somos, Introduction to Ramanujan theta functions %H A298931 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions %F A298931 Expansion of q^(-3/2) * eta(q^8)^2 * eta(q^9)^3 / (eta(q^3) * eta(q^4)) in powers of q. %F A298931 Euler transform of a period 72 sequence. %F A298931 A005872(2*n + 3) = 6*a(n). a(3*n) = A298932(n). a(3*n + 1) = A263452(n-1). a(3*n + 2) = a(4*n + 1) = 0. %e A298931 G.f. = q^3 + q^9 + q^11 + 2*q^15 + q^17 + 2*q^23 + 3*q^27 + 2*q^33 + ... %e A298931 G.f. = 1 + x^3 + x^4 + 2*x^6 + x^7 + 2*x^10 + 3*x^12 + 2*x^15 + ... %t A298931 a[ n_] := SeriesCoefficient[ QPochhammer[ x^8]^2 QPochhammer[ x^9]^3 / (QPochhammer[ x^3] QPochhammer[ x^4]), {x, 0, n}]; %o A298931 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^8 + A)^2 * eta(x^9 + A)^3 / (eta(x^3 + A) * eta(x^4 + A)), n))}; %Y A298931 Cf. A005872, A263452, A298932. %K A298931 nonn,new %O A298931 0,7 %A A298931 _Michael Somos_, Jan 29 2018 %I A298733 %S A298733 1,0,3,-2,9,-6,21,-18,48,-44,99,-102,204,-216,393,-438,747,-846,1362, %T A298733 -1584,2448,-2872,4275,-5082,7356,-8784,12390,-14894,20592,-24798, %U A298733 33651,-40644,54336,-65640,86535,-104628,136356,-164736,212388,-256498,327690,-395214 %N A298733 Expansion of phi(-x^9) * f(-x^3)^2 / f(-x^2)^3 in powers of x where f(), phi() are Ramanujan theta functions. %C A298733 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). %H A298733 M. Somos, Introduction to Ramanujan theta functions %H A298733 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions %F A298733 Expansion of eta(q^3)^2 * eta(q^9)^2 / (eta(q^2)^3 * eta(q^18)) in powers of q. %F A298733 Euler transform of period 18 sequence [0, 3, -2, 3, 0, 1, 0, 3, -4, 3, 0, 1, 0, 3, -2, 3, 0, 0, ...]. %F A298733 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 4/9 g(t) where q = exp(2 pi i t) and g(t) is the g.f. for A182036. %e A298733 G.f. = 1 + 3*x^2 - 2*x^3 + 9*x^4 - 6*x^5 + 21*x^6 - 18*x^7 + 48*x^8 + ... %t A298733 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^9] QPochhammer[ x^3]^2 / QPochhammer[ x^2]^3, {x, 0, n}]; %o A298733 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^9 + A)^2 / (eta(x^2 + A)^3 * eta(x^18 + A)), n))}; %Y A298733 Cf. A182036. %K A298733 sign,new %O A298733 0,3 %A A298733 _Michael Somos_, Jan 29 2018 %I A298883 %S A298883 1,2,6,180,50400,958003200,131514679296000,1352181326649753600000, %T A298883 112703642894318944282214400000, %U A298883 903025586371469323704949549301760000000,2012769637740033870687308804001121075357286400000000 %N A298883 Determinant of n X n matrix whose elements are m(i,j) = prime(i)^j. %C A298883 Traces of these matrices are A087480. %H A298883 Robert Israel, Table of n, a(n) for n = 0..35 %H A298883 Wikipedia, Vandermonde matrix %F A298883 a(n) = Product_{1<=i<=n} prime(i) * Product_{1<=i Determinant(Matrix(n, (i,j)-> ithprime(i)^j)): %p A298883 seq(a(n), n=0..12); # _Alois P. Heinz_, Jan 28 2018 %p A298883 # Alternative: %p A298883 f:= proc(n) local P; %p A298883 P:= [seq(ithprime(i),i=1..n)]; %p A298883 convert(P,`*`)*mul(mul(P[j]-P[i],j=i+1..n),i=1..n-1) %p A298883 end proc: %p A298883 map(f, [$0..20]); # _Robert Israel_, Jan 29 2018 %t A298883 a[n_]:=Table[Prime[i]^j,{i,1,n},{j,1,n}]; %t A298883 Table[Det[a[n]],{n,1,10}] %o A298883 (PARI) a(n) = matdet(matrix(n, n, i, j, prime(i)^j)); \\ _Michel Marcus_, Jan 28 2018 %Y A298883 Cf. A080358, A087480, A298903. %K A298883 nonn,new %O A298883 0,2 %A A298883 _Andres Cicuttin_, Jan 28 2018 %I A298903 %S A298903 1,1,-2,4,48,192,-1216,4672,120704,115712,-1717760,-4103168,10545152, %T A298903 8527872,-520617984,-8178925568,97259454464,-1335459315712, %U A298903 -19462172966912,-360902649708544,-1350652745744384,74944810429972480,-12488535009247887360,-107854339949694287872,84090212651516146221056 %N A298903 Determinant of n X n matrix whose elements are m(i,j) = prime(i+j)-prime(i). %e A298903 For n=1: %e A298903 |prime(2) - prime(1)| = |3 - 2| = |1| = 1, %e A298903 then a(1) = 1. %e A298903 For n=2: %e A298903 |prime(2)-prime(1) prime(3)-prime(1)| = |3-2 5-2| = |1 3|= -2, %e A298903 |prime(3)-prime(2) prime(4)-prime(2)| |5-3 7-3| |2 4| %e A298903 then a(2) = -2. %p A298903 with(LinearAlgebra): %p A298903 a:= n-> Determinant(Matrix(n, (i,j)-> ithprime(i+j)-ithprime(i))): %p A298903 seq(a(n), n=0..25); # _Alois P. Heinz_, Jan 28 2018 %t A298903 b[n_]:=Table[(Prime[i+j]-Prime[i]),{i,1,n},{j,1,n}]; %t A298903 Table[Det[b[n]],{n,1,24}] %o A298903 (PARI) a(n) = matdet(matrix(n, n, i, j, prime(i+j)-prime(i))); \\ _Michel Marcus_, Jan 28 2018 %Y A298903 Cf. A187011, A298883. %K A298903 sign,new %O A298903 0,3 %A A298903 _Andres Cicuttin_, Jan 28 2018 %I A298670 %S A298670 4,5,6,7,8,11,12,13,14,17,18,20,21,23,25,27,28,29,31,32 %N A298670 Numbers n such that A298669(n)*2^n + 1 is a prime factor of the Fermat number 2^(2^(n - 2)) + 1. %Y A298670 Cf. A298669. %K A298670 nonn,more,new %O A298670 1,1 %A A298670 _Arkadiusz Wesolowski_, Jan 24 2018 %I A298669 %S A298669 0,0,1,8,1024,5,1071,6443,52743,1184,11131,39,7,856079,3363658,9264, %T A298669 3150,1313151,13,33629,555296667,534689,8388607,5,512212693,193652, %U A298669 286330,282030,7224372579,1120049,149041 %N A298669 Let b(n) = 2^n with n >= 2, and let c = k*b(n) + 1 for k >= 1; then a(n) is the smallest k such that c is prime and such that A007814(r(n)) = A007814(k) + n where r(n) is the remainder of 2^(b(n)/4) mod c, or 0 if no such k exists. %C A298669 a(n-2) <= A007117(n). %C A298669 a(33) <= 5463561471303. %H A298669 Wilfrid Keller, Fermat factoring status %F A298669 For n >= 1, a(A204620(n)) = 3; a(A226366(n)) = 5; a(A280003(n)) = 7. %o A298669 (PARI) print1(0, ", "0", "); for(n=4, 32, b=2^n; k=1; t=0; while(t<1, c=k*b+1; if(isprime(c), r=Mod(2, c)^(b/4); if(lift(r/b)<=k, if(valuation(lift(r), 2)==valuation(k, 2)+n, t=1; print1(k, ", ")))); k++)); %Y A298669 Cf. A007117, A007814, A298670. %K A298669 nonn,new %O A298669 2,4 %A A298669 _Arkadiusz Wesolowski_, Jan 24 2018 %I A297625 %S A297625 5,17,73,257,65537,262657,4432676798593 %N A297625 Primes of the form (2^(p^k) - 1)/(2^(p^(k - 1)) - 1), with p prime and k > 1. %C A297625 Primes of the form Phi(x, 2), where x is a composite prime power and Phi is the cyclotomic polynomial. %C A297625 Together with 3, supersequence of A019434. %C A297625 Also called Mersenne-Fermat primes. %C A297625 a(8) has 1031 digits and is too large to include. %D A297625 Fredrick Kennard, Unsolved Problems in Mathematics, Lulu Press, 2015, p. 160. %H A297625 Wikipedia, Mersenne-Fermat primes %o A297625 (MAGMA) lst:=[]; r:=7; pr:=PrimesUpTo(r); for k in [2..r] do for c in [1..#pr] do p:=pr[c]; if p^k le r^2 then MF:=Truncate((2^(p^k)-1)/(2^(p^(k-1))-1)); if IsPrime(MF) then Append(~lst, MF); end if; end if; end for; end for; Sort(lst); %Y A297625 Cf. A019434, A051156, A140797, A187823, A245730. %K A297625 nonn,new %O A297625 1,1 %A A297625 _Arkadiusz Wesolowski_, Jan 04 2018 %I A284708 %S A284708 2,2,3,11,37,107,409,409,409,25471,53173,65003,766439,11797483 %N A284708 Smallest initial prime p for n primes in increasing arithmetic progression with a common difference less than p. %C A284708 Conjecture: a(n) > A034386(n) for every n >= 4. %e A284708 Smallest initial prime p, primes in arithmetic progression: %e A284708 a(1) = 2: (2); %e A284708 a(2) = 2: (2, 3); %e A284708 a(3) = 3: (3, 5, 7); %e A284708 a(4) = 11: (11, 17, 23, 29); %e A284708 a(5) = 37: (37, 67, 97, 127, 157); %e A284708 a(6) = 107: (107, 137, 167, 197, 227, 257); %K A284708 nonn,more,new %O A284708 1,1 %A A284708 _Arkadiusz Wesolowski_, Jan 09 2018 %I A298827 %S A298827 4,5,28,41,84,336,990,193,1260,5905,75918,10065,318860,2391485, %T A298827 14348908,20390382,5031420,31624326,5985168,1743333144,8569036, %U A298827 668070480,547062516,141214768241,167874004756,1270932914165,385131186110,2837770056420,784347169884,475536631360,149093578413164,139370386996590 %N A298827 a(n) is the smallest positive integer k such that 3^n+2 divides 3^(n+k)+2. %C A298827 3^n+2 divides 3^(n+a(n)*m)+2 for all nonnegative integers m. %C A298827 Jon E. Schoenfield noted that a(n) coincides with the multiplicative order of 3 modulo 3^n+2. This is true because 3^(n+a(n)) == 3^n mod 3^n+2 and since 3^n and 3^n+2 are coprime, 3^a(n) == 1 mod 3^n+2 and the multiplicative order is the smallest positive such number. - _Chai Wah Wu_, Jan 29 2018 %H A298827 Robert Israel, Table of n, a(n) for n = 1..175 %e A298827 For n = 1, f(1) = 3^1 + 2 = 5, where f(x) = 3^x + 2. Given the last digits of f(x) form a recurring sequence of 5, 1, 9, 3 [, 5, 1, 9, 3] then whenever x = 1 mod 4, f(x) will be a multiple of f(1). %e A298827 For n = 2, f(2) = 3^2 + 2 = 11. a(2) = 5. So any x = 2 mod 5 will be a multiple of 11. For instance, 27 = 2 mod 5, and f(27) = 3^27 + 2 = 7625597474989 = 11 * 693236134999. %p A298827 seq(numtheory:-order(3, 3^n+2), n=1..100); # _Robert Israel_, Feb 05 2018 %t A298827 Array[Block[{k = 1}, While[! Divisible[3^(# + k) + 2, 3^# + 2], k++]; k] &, 12] (* _Michael De Vlieger_, Feb 05 2018 *) %t A298827 Table[MultiplicativeOrder[3, 3^n + 2], {n, 32}] (* _Jean-François Alcover_, Feb 06 2018 *) %o A298827 (Python) %o A298827 def fmod(n, mod): %o A298827 ....return (pow(3, n, mod) + 2) % mod %o A298827 def f(n): %o A298827 ....return pow(3, n) + 2 %o A298827 #terms is the number of terms to generate %o A298827 terms = 20 %o A298827 for x in range(1,terms + 1): %o A298827 ....div = f(x) %o A298827 ....y = x + 1 %o A298827 ....while fmod(y, div) != 0: %o A298827 ........y += 1 %o A298827 ....print(y - x) %o A298827 (Python) %o A298827 from sympy import n_order %o A298827 def A298827(n): %o A298827 return n_order(3,3**n+2) # _Chai Wah Wu_, Jan 29 2018 %o A298827 (MAGMA) [Modorder(3,3^n+2): n in [1..29]]; // _Jon E. Schoenfield_, Jan 28 2018 %o A298827 (PARI) a(n) = znorder(Mod(3, 3^n+2)); \\ _Michel Marcus_, Jan 29 2018 %Y A298827 Cf. A168607. %K A298827 nonn,new %O A298827 1,1 %A A298827 _Luke W. Richards_, Jan 27 2018 %E A298827 a(22)-a(32) from _Robert Israel_, Feb 05 2018 %I A298930 %S A298930 0,0,0,0,1,0,0,3,3,0,0,2,1,2,0,0,11,4,4,11,0,0,13,3,11,3,13,0,0,34,7, %T A298930 23,23,7,34,0,0,65,14,72,94,72,14,65,0,0,123,35,201,255,255,201,35, %U A298930 123,0,0,266,89,597,666,955,666,597,89,266,0,0,499,242,1717,2720,3569,3569,2720 %N A298930 T(n,k)=Number of nXk 0..1 arrays with every element equal to 2, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298930 Table starts %C A298930 .0...0..0....0....0.....0......0.......0........0.........0.........0 %C A298930 .0...1..3....2...11....13.....34......65......123.......266.......499 %C A298930 .0...3..1....4....3.....7.....14......35.......89.......242.......643 %C A298930 .0...2..4...11...23....72....201.....597.....1717......5183.....15479 %C A298930 .0..11..3...23...94...255....666....2720.....8571.....30093....106192 %C A298930 .0..13..7...72..255...955...3569...15031....61046....253624...1067751 %C A298930 .0..34.14..201..666..3569..15163...77576...375845...1886321...9434661 %C A298930 .0..65.35..597.2720.15031..77576..491338..2806944..16889645.100996053 %C A298930 .0.123.89.1717.8571.61046.375845.2806944.19329083.138482223.988954778 %H A298930 R. H. Hardin, Table of n, a(n) for n = 1..220 %F A298930 Empirical for column k: %F A298930 k=1: a(n) = a(n-1) %F A298930 k=2: a(n) = a(n-1) +3*a(n-2) -4*a(n-4) %F A298930 k=3: [order 18] for n>19 %F A298930 k=4: [order 60] for n>61 %e A298930 Some solutions for n=5 k=4 %e A298930 ..0..0..1..1. .0..0..0..0. .0..0..0..0. .0..0..1..1. .0..0..1..1 %e A298930 ..1..0..1..0. .0..0..0..0. .0..1..1..0. .0..1..0..1. .0..1..0..1 %e A298930 ..1..1..0..0. .0..0..0..0. .1..0..1..0. .1..0..0..1. .1..1..0..0 %e A298930 ..1..0..1..0. .1..1..1..1. .1..0..1..0. .0..1..0..1. .0..1..0..1 %e A298930 ..0..0..1..1. .1..1..1..1. .1..1..0..0. .0..0..1..1. .0..0..1..1 %Y A298930 Column 2 is A297870. %Y A298930 Column 3 is A298254. %K A298930 nonn,tabl,new %O A298930 1,8 %A A298930 _R. H. Hardin_, Jan 29 2018 %I A298929 %S A298929 0,34,14,201,666,3569,15163,77576,375845,1886321,9434661,47480566, %T A298929 237250519,1191569971,5975706525,29984756274,150481729037, %U A298929 755439602693,3791563019201,19033524132770,95544814021035,479625376201987 %N A298929 Number of nX7 0..1 arrays with every element equal to 2, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298929 Column 7 of A298930. %H A298929 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298929 Some solutions for n=5 %e A298929 ..0..0..0..1..1..1..1. .0..0..0..1..1..0..0. .0..0..1..0..0..0..0 %e A298929 ..0..0..0..1..0..0..1. .0..0..0..1..0..1..0. .0..1..1..0..0..0..0 %e A298929 ..0..0..0..1..0..0..1. .0..0..0..1..0..0..1. .1..0..0..1..0..0..0 %e A298929 ..0..0..0..1..0..0..1. .0..0..0..1..0..1..0. .0..1..1..0..0..0..0 %e A298929 ..0..0..0..1..1..1..1. .0..0..0..1..1..0..0. .0..0..1..0..0..0..0 %Y A298929 Cf. A298930. %K A298929 nonn,new %O A298929 1,2 %A A298929 _R. H. Hardin_, Jan 29 2018 %I A298928 %S A298928 0,13,7,72,255,955,3569,15031,61046,253624,1067751,4492809,18851449, %T A298928 79305038,333393642,1401954916,5897152044,24806594547,104353275618, %U A298928 439007735235,1846879331358,7769846838226,32688195410797,137521855214717 %N A298928 Number of nX6 0..1 arrays with every element equal to 2, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298928 Column 6 of A298930. %H A298928 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298928 Some solutions for n=5 %e A298928 ..0..0..0..0..1..1. .0..0..0..1..1..1. .0..0..1..1..1..1. .0..0..0..0..0..0 %e A298928 ..0..0..0..0..1..1. .0..0..0..1..1..1. .0..1..0..0..0..1. .0..0..0..0..0..0 %e A298928 ..0..0..0..0..1..1. .0..0..0..1..1..1. .1..0..0..0..0..1. .0..0..0..0..0..0 %e A298928 ..0..0..0..0..1..1. .0..0..0..1..1..1. .1..0..0..0..0..1. .0..0..0..0..0..0 %e A298928 ..0..0..0..0..1..1. .0..0..0..1..1..1. .1..1..1..1..1..1. .0..0..0..0..0..0 %Y A298928 Cf. A298930. %K A298928 nonn,new %O A298928 1,2 %A A298928 _R. H. Hardin_, Jan 29 2018 %I A298927 %S A298927 0,11,3,23,94,255,666,2720,8571,30093,106192,369318,1291934,4552705, %T A298927 15952349,56078262,197277132,693611434,2438933680,8579643205, %U A298927 30173988847,106130105150,373307026724,1313049614295,4618497134678,16245278266932 %N A298927 Number of nX5 0..1 arrays with every element equal to 2, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298927 Column 5 of A298930. %H A298927 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298927 Some solutions for n=5 %e A298927 ..0..0..1..0..0. .0..0..1..0..0. .0..0..1..0..0. .0..0..1..1..1 %e A298927 ..0..1..1..0..0. .0..1..1..1..0. .1..0..1..1..0. .0..0..1..1..1 %e A298927 ..1..1..1..1..1. .1..0..0..0..1. .1..1..0..0..1. .0..0..1..1..1 %e A298927 ..0..0..1..0..1. .0..1..1..1..0. .1..0..1..1..0. .0..0..1..1..1 %e A298927 ..0..1..1..0..0. .0..0..1..0..0. .0..0..1..0..0. .0..0..1..1..1 %Y A298927 Cf. A298930. %K A298927 nonn,new %O A298927 1,2 %A A298927 _R. H. Hardin_, Jan 29 2018 %I A298926 %S A298926 0,2,4,11,23,72,201,597,1717,5183,15479,46260,138928,417427,1255369, %T A298926 3777004,11372190,34247960,103164581,310821235,936579808,2822417917, %U A298926 8506019597,25636254701,77267796877,232891865309,701970354408 %N A298926 Number of nX4 0..1 arrays with every element equal to 2, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298926 Column 4 of A298930. %H A298926 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298926 Empirical: a(n) = 6*a(n-1) -10*a(n-2) +7*a(n-3) -23*a(n-4) +31*a(n-5) -15*a(n-6) +104*a(n-7) -147*a(n-8) +182*a(n-9) -366*a(n-10) +376*a(n-11) -685*a(n-12) +689*a(n-13) -613*a(n-14) +1143*a(n-15) -710*a(n-16) -129*a(n-17) -82*a(n-18) -584*a(n-19) +5620*a(n-20) -1356*a(n-21) +6298*a(n-22) -15575*a(n-23) +6568*a(n-24) -15985*a(n-25) +6661*a(n-26) -23488*a(n-27) +19556*a(n-28) +16109*a(n-29) +18333*a(n-30) +8059*a(n-31) -12197*a(n-32) +16533*a(n-33) -27875*a(n-34) -8576*a(n-35) -28209*a(n-36) +16299*a(n-37) +26487*a(n-38) +4040*a(n-39) -22226*a(n-40) +1134*a(n-41) -9128*a(n-42) -7294*a(n-43) +16692*a(n-44) +6504*a(n-45) +1137*a(n-46) -5253*a(n-47) +2127*a(n-48) -324*a(n-49) +1885*a(n-50) -1689*a(n-51) -59*a(n-52) -235*a(n-53) +346*a(n-54) -86*a(n-55) +12*a(n-56) -46*a(n-57) +24*a(n-58) +8*a(n-59) -4*a(n-60) for n>61 %e A298926 Some solutions for n=5 %e A298926 ..0..0..1..1. .0..0..0..0. .0..0..0..0. .0..0..1..1. .0..0..0..0 %e A298926 ..0..1..0..1. .0..1..1..0. .0..0..0..0. .0..1..0..1. .0..1..1..0 %e A298926 ..1..1..0..1. .1..0..1..0. .0..0..0..0. .0..1..0..1. .0..1..0..1 %e A298926 ..0..1..0..1. .1..0..1..0. .1..1..1..1. .1..0..0..1. .1..0..0..1 %e A298926 ..0..0..1..1. .1..1..0..0. .1..1..1..1. .1..1..1..1. .1..1..1..1 %Y A298926 Cf. A298930. %K A298926 nonn,new %O A298926 1,2 %A A298926 _R. H. Hardin_, Jan 29 2018 %I A298925 %S A298925 0,1,1,11,94,955,15163,491338,19329083,1185260593 %N A298925 Number of nXn 0..1 arrays with every element equal to 2, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298925 Diagonal of A298930. %e A298925 Some solutions for n=5 %e A298925 ..0..0..1..0..0. .0..0..1..0..0. .0..0..1..0..0. .0..0..1..0..0 %e A298925 ..0..1..0..1..0. .0..1..0..1..0. .1..0..1..1..0. .0..1..1..0..1 %e A298925 ..1..1..0..1..0. .0..1..1..1..0. .1..1..0..0..1. .1..0..0..1..1 %e A298925 ..0..1..0..1..0. .0..1..0..0..1. .0..1..1..1..0. .1..0..1..1..0 %e A298925 ..0..0..1..0..0. .0..0..1..1..1. .0..0..0..0..0. .1..1..0..0..0 %Y A298925 Cf. A298930. %K A298925 nonn,new %O A298925 1,4 %A A298925 _R. H. Hardin_, Jan 29 2018 %I A298924 %S A298924 0,1,1,0,4,0,1,4,4,1,0,8,1,8,0,1,32,2,2,32,1,0,32,3,3,3,32,0,1,64,5,7, %T A298924 7,5,64,1,0,256,8,6,9,6,8,256,0,1,256,16,9,12,12,9,16,256,1,0,512,21, %U A298924 20,17,19,17,20,21,512,0,1,2048,34,22,41,22,22,41,22,34,2048,1,0,2048,55,35,42 %N A298924 T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298924 Table starts %C A298924 .0...1..0..1..0..1..0...1...0...1....0....1....0.....1.....0.....1......0 %C A298924 .1...4..4..8.32.32.64.256.256.512.2048.2048.4096.16384.16384.32768.131072 %C A298924 .0...4..1..2..3..5..8..16..21..34...55...89..144...236...377...610....987 %C A298924 .1...8..2..3..7..6..9..20..22..35...59...90..145...240...378...611....991 %C A298924 .0..32..3..7..9.12.17..41..42..67..109..172..277...461...722..1167...1889 %C A298924 .1..32..5..6.12.19.22..48..53.103..169..272..446...863..1346..2395...4154 %C A298924 .0..64..8..9.17.22.31..83..92.172..309..549..923..1830..3021..5580..10091 %C A298924 .1.256.16.20.41.48.83.199.225.460..943.1570.2795..5933.10113.19462..37621 %C A298924 .0.256.21.22.42.53.92.225.330.644.1348.2306.4339..9315.17574.35170..70446 %H A298924 R. H. Hardin, Table of n, a(n) for n = 1..543 %F A298924 Empirical for column k: %F A298924 k=1: a(n) = a(n-2) %F A298924 k=2: a(n) = 8*a(n-3) %F A298924 k=3: a(n) = a(n-1) +a(n-2) +a(n-6) -a(n-7) -a(n-8) %F A298924 k=4: a(n) = a(n-1) +a(n-2) +a(n-6) -a(n-7) -a(n-8) %F A298924 k=5: a(n) = a(n-1) +a(n-2) +a(n-6) -a(n-7) -a(n-8) for n>11 %e A298924 Some solutions for n=5 k=4 %e A298924 ..0..0..0..0. .0..0..1..1. .0..1..0..0. .0..0..0..0. .0..0..1..0 %e A298924 ..0..0..0..0. .0..0..1..1. .0..1..0..0. .0..0..0..0. .0..0..1..0 %e A298924 ..0..0..0..0. .0..0..1..1. .1..1..0..0. .0..0..0..0. .0..0..1..1 %e A298924 ..1..1..1..1. .0..0..1..1. .0..1..0..0. .0..0..0..0. .0..0..1..0 %e A298924 ..1..1..1..1. .0..0..1..1. .0..1..0..0. .0..0..0..0. .0..0..1..0 %Y A298924 Column 2 is A096252(n-1). %K A298924 nonn,tabl,new %O A298924 1,5 %A A298924 _R. H. Hardin_, Jan 29 2018 %I A298923 %S A298923 0,64,8,9,17,22,31,83,92,172,309,549,923,1830,3021,5580,10091,18344, %T A298923 33025,61470,111014,205198,376471,694579,1277419,2365958,4360085, %U A298923 8074209,14929946,27656016,51194508,94924809,175864241,326176032,604808134 %N A298923 Number of nX7 0..1 arrays with every element equal to 1, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298923 Column 7 of A298924. %H A298923 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298923 Some solutions for n=5 %e A298923 ..0..0..0..0..0..1..1. .0..0..0..0..1..1..1. .0..0..0..1..1..0..1 %e A298923 ..0..0..0..0..0..1..1. .0..0..0..0..1..1..1. .0..0..0..1..1..0..1 %e A298923 ..0..0..0..0..0..1..1. .0..0..0..0..1..1..1. .0..0..0..1..1..0..0 %e A298923 ..0..0..0..0..0..1..1. .0..0..0..0..1..1..1. .0..0..0..1..1..0..1 %e A298923 ..0..0..0..0..0..1..1. .0..0..0..0..1..1..1. .0..0..0..1..1..0..1 %Y A298923 Cf. A298924. %K A298923 nonn,new %O A298923 1,2 %A A298923 _R. H. Hardin_, Jan 29 2018 %I A298922 %S A298922 1,32,5,6,12,19,22,48,53,103,169,272,446,863,1346,2395,4154,7334, %T A298922 12706,22695,39500,70143,123441,218719,385766,684533,1208762,2142932, %U A298922 3791443,6719743,11895717,21085505,37340353,66178799,117235756,207768546 %N A298922 Number of nX6 0..1 arrays with every element equal to 1, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298922 Column 6 of A298924. %H A298922 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298922 Some solutions for n=5 %e A298922 ..0..0..1..1..0..1. .0..0..0..0..1..0. .0..0..0..1..1..1. .0..0..0..0..0..0 %e A298922 ..0..0..1..1..0..1. .0..0..0..0..1..0. .0..0..0..1..1..1. .0..0..0..0..0..0 %e A298922 ..0..0..1..1..0..0. .0..0..0..0..1..1. .0..0..0..1..1..1. .1..1..1..1..1..1 %e A298922 ..0..0..1..1..0..1. .0..0..0..0..1..0. .0..0..0..1..1..1. .1..1..1..1..1..1 %e A298922 ..0..0..1..1..0..1. .0..0..0..0..1..0. .0..0..0..1..1..1. .1..1..1..1..1..1 %Y A298922 Cf. A298924. %K A298922 nonn,new %O A298922 1,2 %A A298922 _R. H. Hardin_, Jan 29 2018 %I A298921 %S A298921 0,32,3,7,9,12,17,41,42,67,109,172,277,461,722,1167,1889,3052,4937, %T A298921 8001,12922,20907,33829,54732,88557,143301,231842,375127,606969, %U A298921 982092,1589057,2571161,4160202,6731347,10891549,17622892,28514437,46137341 %N A298921 Number of nX5 0..1 arrays with every element equal to 1, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298921 Column 5 of A298924. %H A298921 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298921 Empirical: a(n) = a(n-1) +a(n-2) +a(n-6) -a(n-7) -a(n-8) for n>11 %e A298921 Some solutions for n=5 %e A298921 ..0..1..0..0..0. .0..0..0..1..0. .0..0..0..0..0. .0..0..1..1..1 %e A298921 ..0..1..0..0..0. .0..0..0..1..0. .0..0..0..0..0. .0..0..1..1..1 %e A298921 ..1..1..0..0..0. .0..0..0..1..1. .1..1..1..1..1. .0..0..1..1..1 %e A298921 ..0..1..0..0..0. .0..0..0..1..0. .1..1..1..1..1. .0..0..1..1..1 %e A298921 ..0..1..0..0..0. .0..0..0..1..0. .1..1..1..1..1. .0..0..1..1..1 %Y A298921 Cf. A298924. %K A298921 nonn,new %O A298921 1,2 %A A298921 _R. H. Hardin_, Jan 29 2018 %I A298920 %S A298920 1,8,2,3,7,6,9,20,22,35,59,90,145,240,378,611,991,1598,2585,4188,6766, %T A298920 10947,17715,28658,46369,75032,121394,196419,317815,514230,832041, %U A298920 1346276,2178310,3524579,5702891,9227466,14930353,24157824,39088170 %N A298920 Number of nX4 0..1 arrays with every element equal to 1, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298920 Column 4 of A298924. %H A298920 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298920 Empirical: a(n) = a(n-1) +a(n-2) +a(n-6) -a(n-7) -a(n-8) %e A298920 Some solutions for n=5 %e A298920 ..0..0..0..0. .0..1..0..1. .0..0..0..0. .0..0..0..0. .0..0..1..1 %e A298920 ..0..0..0..0. .0..1..0..1. .0..0..0..0. .0..0..0..0. .0..0..1..1 %e A298920 ..1..1..1..1. .1..1..0..0. .0..0..0..0. .0..0..0..0. .0..0..1..1 %e A298920 ..1..1..1..1. .0..1..0..1. .1..1..1..1. .0..0..0..0. .0..0..1..1 %e A298920 ..1..1..1..1. .0..1..0..1. .1..1..1..1. .0..0..0..0. .0..0..1..1 %Y A298920 Cf. A298924. %K A298920 nonn,new %O A298920 1,2 %A A298920 _R. H. Hardin_, Jan 29 2018 %I A298919 %S A298919 0,4,1,2,3,5,8,16,21,34,55,89,144,236,377,610,987,1597,2584,4184,6765, %T A298919 10946,17711,28657,46368,75028,121393,196418,317811,514229,832040, %U A298919 1346272,2178309,3524578,5702887,9227465,14930352,24157820,39088169 %N A298919 Number of nX3 0..1 arrays with every element equal to 1, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298919 Column 3 of A298924. %H A298919 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298919 Empirical: a(n) = a(n-1) +a(n-2) +a(n-6) -a(n-7) -a(n-8) %e A298919 All solutions for n=5 %e A298919 ..0..0..0. .0..0..0. .0..0..0 %e A298919 ..0..0..0. .0..0..0. .0..0..0 %e A298919 ..0..0..0. .0..0..0. .1..1..1 %e A298919 ..1..1..1. .0..0..0. .1..1..1 %e A298919 ..1..1..1. .0..0..0. .1..1..1 %Y A298919 Cf. A298924. %K A298919 nonn,new %O A298919 1,2 %A A298919 _R. H. Hardin_, Jan 29 2018 %I A298918 %S A298918 0,4,1,3,9,19,31,199,330,1377,6627,23223,113179,849856,5207476, %T A298918 43357615 %N A298918 Number of nXn 0..1 arrays with every element equal to 1, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298918 Diagonal of A298924. %e A298918 Some solutions for n=5 %e A298918 ..0..1..0..0..0. .0..0..0..0..0. .0..0..0..0..0. .0..0..1..1..1 %e A298918 ..0..1..0..0..0. .0..0..0..0..0. .0..0..0..0..0. .0..0..1..1..1 %e A298918 ..1..1..0..0..0. .0..0..0..0..0. .0..0..0..0..0. .0..0..1..1..1 %e A298918 ..0..1..0..0..0. .0..0..0..0..0. .1..1..1..1..1. .0..0..1..1..1 %e A298918 ..0..1..0..0..0. .0..0..0..0..0. .0..0..1..0..0. .0..0..1..1..1 %Y A298918 Cf. A298924. %K A298918 nonn,new %O A298918 1,2 %A A298918 _R. H. Hardin_, Jan 29 2018 %I A298563 %S A298563 1,3,5,6,14,44,110,152,884,2144,8384,18632,116624,8394752,15370304, %T A298563 73995392,536920064,2147581952,34360131584 %N A298563 Numbers k such that k - 2 | sigma(k). %C A298563 Similar to A055708. %C A298563 Sequence includes every number of the form 2^(j-1)*(2^j+3) such that 2^j+3 is prime (i.e., j is a term in A057732); terms of this form are 5, 14, 44, 152, 2144, 8384, 8394752, 536920064, 2147581952, 34360131584, ... - _Jon E. Schoenfield_, Jan 22 2018 %C A298563 Superset of A125246. a(20) > 10^12. - _Giovanni Resta_, Jan 23 2018 %e A298563 For k=44, sigma(k)/(k-2) = sigma(44)/(44-2) = 84/42 = 2, so 44 belongs to the sequence; %e A298563 for k=110, sigma(k)/(k-2) = sigma(110)/(110-2) = 216/108 = 2, so 110 is also a term. %t A298563 Select[Range[10^6], Divisible[DivisorSigma[1, #], # - 2] &] (* _Michael De Vlieger_, Jan 21 2018 *) %o A298563 (PARI) isok(k) = (k!=2) && !(sigma(k) % (k-2)); \\ _Michel Marcus_, Jan 22 2018 %o A298563 (MAGMA) [n: n in [3..10^7]| DivisorSigma(1, n) mod (n-2) eq 0]; // _Vincenzo Librandi_, Jan 22 2018 %Y A298563 Cf. A055708, A057732, A125246. %K A298563 nonn,more,new %O A298563 1,2 %A A298563 _Zdenek Cervenka_, Jan 21 2018 %E A298563 a(17)-a(18) from _Robert G. Wilson v_, Jan 21 2018 %E A298563 a(19) from _Giovanni Resta_, Jan 23 2018 %I A298917 %S A298917 1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,3,2,2,3,1,1,5,3,3,3,5,1,1,8,5,4,4,5, %T A298917 8,1,1,13,8,6,7,6,8,13,1,1,21,13,9,9,9,9,13,21,1,1,34,21,14,15,14,15, %U A298917 14,21,34,1,1,55,34,22,26,24,24,26,22,34,55,1,1,89,55,35,46,44,40,44,46,35,55 %N A298917 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298917 Table starts %C A298917 .1..1..1..1..1..1...1...1...1....1....1.....1.....1......1......1.......1 %C A298917 .1..1..1..2..3..5...8..13..21...34...55....89...144....233....377.....610 %C A298917 .1..1..1..2..3..5...8..13..21...34...55....89...144....233....377.....610 %C A298917 .1..2..2..3..4..6...9..14..22...35...56....90...145....234....378.....611 %C A298917 .1..3..3..4..7..9..15..26..46...84..151...276...506....929...1708....3138 %C A298917 .1..5..5..6..9.14..24..44..81..156..306...602..1192...2370...4720....9415 %C A298917 .1..8..8..9.15.24..40..76.141..277..570..1171..2441...5157..10913...23193 %C A298917 .1.13.13.14.26.44..76.168.359..792.1895..4521.10886..26818..66131..163463 %C A298917 .1.21.21.22.46.81.141.359.873.2145.5971.16568.45898.131372.376833.1078872 %H A298917 R. H. Hardin, Table of n, a(n) for n = 1..612 %F A298917 Empirical for column k: %F A298917 k=1: a(n) = a(n-1) %F A298917 k=2: a(n) = a(n-1) +a(n-2) for n>3 %F A298917 k=3: a(n) = a(n-1) +a(n-2) for n>3 %F A298917 k=4: a(n) = 2*a(n-1) -a(n-3) %F A298917 k=5: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-5) -a(n-6) -a(n-7) -a(n-8) for n>9 %F A298917 k=6: a(n) = 3*a(n-1) -2*a(n-2) +a(n-3) -2*a(n-4) -a(n-7) +2*a(n-8) for n>9 %F A298917 k=7: [order 14] for n>15 %e A298917 All solutions for n=5 k=4 %e A298917 ..0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..0..0 %e A298917 ..0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..0..0 %e A298917 ..1..1..1..1. .0..0..1..1. .0..0..0..0. .0..0..0..0 %e A298917 ..1..1..1..1. .0..0..1..1. .1..1..1..1. .0..0..0..0 %e A298917 ..1..1..1..1. .0..0..1..1. .1..1..1..1. .0..0..0..0 %Y A298917 Columns 2 and 3 are A000045(n-1). %Y A298917 Column 4 is A001611(n-1). %K A298917 nonn,tabl,new %O A298917 1,12 %A A298917 _R. H. Hardin_, Jan 29 2018 %I A298916 %S A298916 1,8,8,9,15,24,40,76,141,277,570,1171,2441,5157,10913,23193,49468, %T A298916 105598,225691,482849,1033370,2212340,4737746,10147163,21735151, %U A298916 46560372,99744293,213684681,457793103,980778481,2101245689,4501796828,9644879742 %N A298916 Number of nX7 0..1 arrays with every element equal to 0, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298916 Column 7 of A298917. %H A298916 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298916 Empirical: a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-3) -9*a(n-4) +7*a(n-5) -2*a(n-6) -a(n-7) +7*a(n-8) -5*a(n-9) +3*a(n-10) -2*a(n-11) -a(n-12) +a(n-13) -a(n-14) for n>15 %e A298916 Some solutions for n=5 %e A298916 ..0..0..0..0..1..1..1. .0..0..0..0..0..1..1. .0..0..0..0..0..0..0 %e A298916 ..0..0..0..0..1..1..1. .0..0..0..0..0..1..1. .0..0..0..0..0..0..0 %e A298916 ..0..0..0..0..1..1..1. .0..0..0..0..0..1..1. .1..1..1..1..1..1..1 %e A298916 ..0..0..0..0..1..1..1. .0..0..0..0..0..1..1. .1..1..1..1..1..1..1 %e A298916 ..0..0..0..0..1..1..1. .0..0..0..0..0..1..1. .1..1..1..1..1..1..1 %Y A298916 Cf. A298917. %K A298916 nonn,new %O A298916 1,2 %A A298916 _R. H. Hardin_, Jan 29 2018 %I A298915 %S A298915 1,5,5,6,9,14,24,44,81,156,306,602,1192,2370,4720,9415,18797,37547, %T A298915 75032,149981,299846,599534,1198852,2397405,4794400,9588236,19175692, %U A298915 38350310,76699140,153396236,306789653,613575417,1227145465,2454283522,4908556821 %N A298915 Number of nX6 0..1 arrays with every element equal to 0, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298915 Column 6 of A298917. %H A298915 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298915 Empirical: a(n) = 3*a(n-1) -2*a(n-2) +a(n-3) -2*a(n-4) -a(n-7) +2*a(n-8) for n>9 %e A298915 Some solutions for n=5 %e A298915 ..0..0..0..0..0..0. .0..0..0..1..1..1. .0..0..0..0..0..0. .0..0..1..1..0..0 %e A298915 ..0..0..0..0..0..0. .0..0..0..1..1..1. .0..0..0..0..0..0. .0..0..1..1..0..0 %e A298915 ..0..0..0..0..0..0. .0..0..0..1..1..1. .0..0..0..1..0..0. .0..0..1..1..0..0 %e A298915 ..1..1..1..1..1..1. .0..0..0..1..1..1. .0..0..0..0..0..0. .0..0..1..1..0..0 %e A298915 ..1..1..1..1..1..1. .0..0..0..1..1..1. .0..0..0..0..0..0. .0..0..1..1..0..0 %Y A298915 Cf. A298917. %K A298915 nonn,new %O A298915 1,2 %A A298915 _R. H. Hardin_, Jan 29 2018 %I A298914 %S A298914 1,3,3,4,7,9,15,26,46,84,151,276,506,929,1708,3138,5770,10611,19515, %T A298914 35893,66014,121417,223319,410746,755479,1389539,2555759,4700772, %U A298914 8646066,15902594,29249427,53798082,98950098,181997603,334745780,615693476 %N A298914 Number of nX5 0..1 arrays with every element equal to 0, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298914 Column 5 of A298917. %H A298914 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298914 Empirical: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-5) -a(n-6) -a(n-7) -a(n-8) for n>9 %e A298914 Some solutions for n=5 %e A298914 ..0..0..1..1..1. .0..0..0..1..1. .0..0..0..0..0. .0..0..0..0..0 %e A298914 ..0..0..1..1..1. .0..0..0..1..1. .0..0..0..0..0. .0..0..0..0..0 %e A298914 ..0..0..1..1..1. .0..0..0..1..1. .0..0..0..0..0. .0..0..0..0..0 %e A298914 ..0..0..1..1..1. .0..0..0..1..1. .1..1..1..1..1. .0..0..0..0..0 %e A298914 ..0..0..1..1..1. .0..0..0..1..1. .1..1..1..1..1. .0..0..0..0..0 %Y A298914 Cf. A298917. %K A298914 nonn,new %O A298914 1,2 %A A298914 _R. H. Hardin_, Jan 29 2018 %I A298913 %S A298913 1,1,1,3,7,14,40,168,873,5632,63538,1026354,20200595,622554839, %T A298913 29122695476,1747360635695,149225079090933 %N A298913 Number of nXn 0..1 arrays with every element equal to 0, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero. %C A298913 Diagonal of A298917. %e A298913 Some solutions for n=5 %e A298913 ..0..1..0..1..0. .0..0..0..1..1. .0..0..0..0..0. .0..0..1..1..1 %e A298913 ..1..1..1..1..1. .0..0..0..1..1. .0..0..0..0..0. .0..0..1..1..1 %e A298913 ..0..1..1..1..0. .0..0..0..1..1. .1..1..1..1..1. .0..0..1..1..1 %e A298913 ..1..1..1..1..1. .0..0..0..1..1. .1..1..1..1..1. .0..0..1..1..1 %e A298913 ..0..1..0..1..0. .0..0..0..1..1. .1..1..1..1..1. .0..0..1..1..1 %Y A298913 Cf. A298917. %K A298913 nonn,new %O A298913 1,4 %A A298913 _R. H. Hardin_, Jan 29 2018 %I A296028 %S A296028 0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,0,1,1,0,0,0,1,0,0,1,1,0,0,1,0,0, %T A296028 0,1,0,0,0,1,1,0,0,0,1,0,0,1,1,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,1,0,0,1, %U A296028 1,0,0,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,1 %N A296028 Characteristic function of primes in the nonmultiples of 3. %F A296028 From _David A. Corneth_, Dec 03 2017: (Start) %F A296028 a(n) = A010051(A001651(n)). %F A296028 a(n) = 1 if (6n - 3 - (-1)^n)/4 is prime, otherwise a(n) = 0. (End) %e A296028 a(2) = 1 because the 2nd nonmultiple of 3 is 2, which is prime. %p A296028 f:= n -> charfcn[{true}](isprime(floor((3*n-1)/2))): %p A296028 map(f, [$1..1000]); # _Robert Israel_, Jan 24 2018 %t A296028 Array[Boole@ PrimeQ@ Floor[(3 # - 1)/2] &, 105] (* _Michael De Vlieger_, Dec 03 2017 *) %o A296028 (PARI) a(n) = isprime(floor((3*n-1)/2)) \\ _Iain Fox_, Dec 03 2017 %o A296028 (PARI) first(n) = {my(inc = t = 1, res = vector(n)); for(i = 1, n, res[i] = isprime(t); t += inc; inc = 3-inc); res} \\ _David A. Corneth_, Dec 03 2017 %Y A296028 Cf. A001651, A010051, A045344. %K A296028 nonn,new %O A296028 1 %A A296028 _Martin Michael Musatov_, Dec 03 2017 %I A297775 %S A297775 1,1,1,1,1,1,2,1,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,1,2,2, %T A297775 2,2,2,2,2,1,2,2,2,2,2,2,2,1,2,2,3,3,3,3,3,2,1,2,2,2,2,2,3,2,2,3,3,3, %U A297775 3,3,2,3,2,3,3,3,3,2,3,3,2,3,3,3,2,3 %N A297775 Number of distinct runs in base-7 digits of n. %C A297775 Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. %H A297775 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297775 1234567 in base-7: 1,3,3,3,1,2,1,5; six runs, of which 4 are distinct, so that a(1234567) = 4. %t A297775 b = 7; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297775 Table[s[n], {n, 1, 200}] %Y A297775 Cf. A043559 (number of runs, not necessarily distinct), A297770, A043534. %K A297775 nonn,base,easy,new %O A297775 1,7 %A A297775 _Clark Kimberling_, Jan 27 2018 %I A298728 %S A298728 193,194,195,196,197,198,199,200,201,209,210,211,212,213,214,215,216, %T A298728 217,226,227,228,229,230,231,232,233 %N A298728 EBCDIC codes for upper case letters. %C A298728 Extended Binary Coded Decimal Interchange Code. %H A298728 IBM Corp, Code Page 37 %H A298728 Wikipedia, EBCDIC %e A298728 Hexadecimal values: C1, C2, C3, C4, C5, C6, C7, C8, C9, D1, D2, D3, D4, D5, D6, D7, D8, D9, E2, E3, E4, E5, E6, E7, E8, E9. %e A298728 For example, 'A' is C1, 'B' is C2, ..., 'Z' is E9. %t A298728 With[{r = Range@ 9}, Flatten@ {2^7 + 2^6 + {r, 2^4 + r, 2^5 + Rest[r]}}] (* _Michael De Vlieger_, Jan 28 2018 *) %Y A298728 Cf. A121378, A298720. %K A298728 base,fini,full,nonn,word,new %O A298728 1,1 %A A298728 _Yohei Furutono_, Jan 25 2018 %I A295759 %S A295759 1,2,8,50,432,4690,61208,933090,16268640,319249698,6963071784, %T A295759 167093039122,4374954323216,124108887889522,3791902447022648, %U A295759 124138462767883202,4335205955612166848,160865445090615444546,6320573384125953811016,262147404448177963790834,11445191965935999115186288 %N A295759 O.g.f.: Sum_{n>=0} Product_{k=1..n} tan( (2*k)*arctan(x) ). %e A295759 O.g.f: A(x) = 1 + 2*x + 8*x^2 + 50*x^3 + 432*x^4 + 4690*x^5 + 61208*x^6 + 933090*x^7 + 16268640*x^8 + 319249698*x^9 + 6963071784*x^10 + + ... %e A295759 such that %e A295759 A(x) = 1 + tan(2*arctan(x)) + tan(2*arctan(x))*tan(4*arctan(x)) + tan(2*arctan(x))*tan(4*arctan(x))*tan(6*arctan(x)) + tan(2*arctan(x))*tan(4*arctan(x))*tan(6*arctan(x))*tan(8*arctan(x)) + tan(2*arctan(x))*tan(4*arctan(x))*tan(6*arctan(x))*tan(8*arctan(x))*tan(10*arctan(x)) + ... %o A295759 (PARI) {a(n)=local(X=x+x*O(x^n), Gf); Gf=sum(m=0, n, prod(k=1, m, tan((2*k)*atan(X)))); polcoeff(Gf, n)} %o A295759 for(n=0,20,print1(a(n),", ")) %Y A295759 Cf. A177381, A295758. %K A295759 nonn,new %O A295759 0,2 %A A295759 _Paul D. Hanna_, Jan 28 2018 %I A295758 %S A295758 1,1,3,15,113,1105,13219,187103,3058113,56675297,1174295267, %T A295758 26898243439,674916701169,18409502066097,542373965958595, %U A295758 17164148092886207,580677914417571585,20913258579319759041,798876414332323236931,32261582928825038942671,1373304514339211081661169 %N A295758 O.g.f.: Sum_{n>=0} Product_{k=1..n} tan( (2*k-1)*arctan(x) ). %e A295758 O.g.f: A(x) = 1 + x + 3*x^2 + 15*x^3 + 113*x^4 + 1105*x^5 + 13219*x^6 + 187103*x^7 + 3058113*x^8 + 56675297*x^9 + 1174295267*x^10 + ... %e A295758 such that %e A295758 A(x) = 1 + x + x*tan(3*arctan(x)) + x*tan(3*arctan(x))*tan(5*arctan(x)) + x*tan(3*arctan(x))*tan(5*arctan(x))*tan(7*arctan(x)) + x*tan(3*arctan(x))*tan(5*arctan(x))*tan(7*arctan(x))*tan(9*arctan(x)) + ... %o A295758 (PARI) {a(n)=local(X=x+x*O(x^n), Gf); Gf=sum(m=0, n, prod(k=1, m, tan((2*k-1)*atan(X)))); polcoeff(Gf, n)} %o A295758 for(n=0,30,print1(a(n),", ")) %Y A295758 Cf. A177381, A295759. %K A295758 nonn,new %O A295758 0,3 %A A295758 _Paul D. Hanna_, Jan 28 2018 %I A298477 %S A298477 2,2,4,12,24,12,28,4,16,60,4,24,140,2,32,230,1112,36,332,4 %N A298477 a(n) is the number of length n strings over a two-letter alphabet that have a minimum palindromic partition size of A090701(n). %C A298477 A "minimum palindromic partition size" of a string is the fewest number of palindromes that the string can be partitioned into. %C A298477 All terms are even because the letters of the alphabet can be swapped (e.g., "101100" can become "010011".) %e A298477 The a(6) = 12 six-character strings that require A090701(6) = 3 partitions are: %e A298477 100101 via (1001)(0)(1), %e A298477 100110 via (1001)(1)(0), %e A298477 101001 via (1)(0)(1001), %e A298477 101100 via (101)(1)(00), %e A298477 110010 via (1)(1001)(0), %e A298477 110100 via (11)(010)(0), %e A298477 along with the six strings made from swapping the 0s and 1s. %Y A298477 Cf. A090701. %K A298477 nonn,more,new %O A298477 1,1 %A A298477 _Peter Kagey_, Jan 19 2018 %I A298476 %S A298476 1,2,37,203,1332,13428,160884,858740 %N A298476 Least k such that A298475(k) = n. %C A298476 A298474(n) = floor(log_2(a(n))) + 1. %F A298476 a(n) >= 2^(A298474(n) - 1). %F A298476 a(n) < 2^A298474(n). %e A298476 The smallest partition of the binary representation of A298476(k) is: %e A298476 k | A298476(k) | partition %e A298476 --+------------+--------------------------------------- %e A298476 1 | 1 | (1)_2 %e A298476 2 | 2 | (1)(0)_2 %e A298476 3 | 37 | (1001)(0)(1)_2 %e A298476 4 | 203 | (11)(00)(101)(1)_2 %e A298476 5 | 1332 | (101)(00)(1)(101)(00)_2 %e A298476 6 | 13428 | (11)(010)(0)(01110)(1)(00)_2 %e A298476 7 | 160884 | (1001)(1)(101)(000)(111)(010)(0)_2 %e A298476 8 | 858740 | (11)(010)(0)(0110)(1001)(11)(010)(0)_2 %t A298476 With[{s = {1, 2}~Join~Array[Function[w, Min@ Map[Length, Select[#, And[AllTrue[#, PalindromeQ], Union@ Map[Length, #] != {1}] &]] &@ Union@ Map[Select[SplitBy[#, IntegerQ], IntegerQ@ First@ # &] &, Map[Insert[w, ".", #] &, Map[{#} &, Rest@ Subsets@ Range@ Length@ w, {2}]]]]@ IntegerDigits[#, 2] &, 1400, 3]}, Array[FirstPosition[s, #][[1]] &, Max@ Take[#, 1 + LengthWhile[Differences@ #, # == 1 &]] &@ Union@ s]] (* _Michael De Vlieger_, Jan 23 2018 *) %Y A298476 Cf. A298474, A298475. %K A298476 nonn,base,more,new %O A298476 1,2 %A A298476 _Peter Kagey_, Jan 19 2018 %I A298475 %S A298475 1,2,1,2,1,2,1,2,1,2,2,2,2,2,1,2,1,2,2,2,1,2,2,2,2,2,1,2,2,2,1,2,1,2, %T A298475 2,2,3,3,2,2,3,2,2,3,1,2,2,2,2,3,1,3,2,2,2,2,2,2,2,2,2,2,1,2,1,2,2,2, %U A298475 3,3,2,2,1,2,3,2,2,3,2,2,3,2,3,2,1 %N A298475 Minimal size of a palindromic partition of the binary representation of n. %C A298475 A palindromic partition of "xxoxoxxox" is (x)(xoxox)(xox). %H A298475 Peter Kagey, Table of n, a(n) for n = 1..10000 %e A298475 The following table shows the partitions of binary representations of n into a(n) palindromes: %e A298475 n | a(n) | binary | partition %e A298475 ----+------+----------+----------------- %e A298475 5 | 1 | 101 | (101) %e A298475 6 | 2 | 110 | (11)(0) %e A298475 7 | 1 | 111 | (111) %e A298475 13 | 2 | 1101 | (1)(101) %e A298475 37 | 3 | 100101 | (1001)(0)(1) %e A298475 203 | 4 | 11001011 | (11)(00)(101)(1) %t A298475 {1, 2}~Join~Array[Function[w, Min@ Map[Length, Select[#, And[AllTrue[#, PalindromeQ], Union@ Map[Length, #] != {1}] &]] &@ Union@ Map[Select[SplitBy[#, IntegerQ], IntegerQ@ First@ # &] &, Map[Insert[w, ".", #] &, Map[{#} &, Rest@ Subsets@ Range@ Length@ w, {2}]]]]@ IntegerDigits[#, 2] &, 103, 3] (* _Michael De Vlieger_, Jan 23 2018 *) %Y A298475 Cf. A006995, A298474. %K A298475 nonn,base,new %O A298475 1,2 %A A298475 _Peter Kagey_, Jan 19 2018 %I A298474 %S A298474 1,2,6,8,11,14,18,20,24,26,30,32,36,38,42,44,48,50,54,56,60,62,66,68, %T A298474 72,74,78,80,84,86,90,92,96,98,102,104,108,110,114,116,120,122,126, %U A298474 128,132,134,138,140,144,146,150,152,156,158,162,164,168,170,174 %N A298474 a(n) is the least k such that A090701(k) = n. %H A298474 Robert Israel, Table of n, a(n) for n = 1..10000 %F A298474 a(n) = floor(log_2(A298476(n))) + 1. %F A298474 From _Robert Israel_, Jan 24 2018: (Start) %F A298474 If n is even, a(n) = 3*n-4. %F A298474 If n <> 1 or 5 is odd, a(n) = 3*n-3. %F A298474 G.f.: x*(1+x+3*x^2+x^3-x^4+x^5+x^6-x^7)/((1-x)*(1-x^2)). (End) %e A298474 The lexicographically earliest strings of length a(n) with a minimum palindromic partition into n parts: %e A298474 n | a(n) | string | partition %e A298474 --+------+----------------+--------------------------- %e A298474 1 | 1 | 0 | (0) %e A298474 2 | 2 | 01 | (0)(1) %e A298474 3 | 6 | 001011 | (0)(010)(11) %e A298474 4 | 8 | 00101100 | (00)(101)(1)(00) %e A298474 5 | 11 | 00101100101 | (00)(101)(1001)(0)(1) %e A298474 6 | 14 | 00101110001011 | (00)(101)(11)(00)(010)(11) %p A298474 f:= n -> 3*n-4+(n mod 2): %p A298474 f(1):= 1: f(5):= 11: %p A298474 map(f, [$1..100]); # _Robert Israel_, Jan 24 2018 %t A298474 With[{s = Array[Boole[# == 11] + Floor[#/6] + Floor[(# + 4)/6] + 1 &, 2^8]}, Array[FirstPosition[s, #][[1]] &, Max@ Take[#, LengthWhile[Differences@ #, # == 1 &]] &@ Union@ s]] (* _Michael De Vlieger_, Jan 23 2018 *) %Y A298474 Cf. A090701, A298476. %K A298474 nonn,new %O A298474 1,2 %A A298474 _Peter Kagey_, Jan 19 2018 %I A298676 %S A298676 1,2,3,5,5,7,7,10,11,13,13,18,19,26,31,36,41,48,59,71,84,94,106,123, %T A298676 146,165,187,210,240,275,318,364,407,465,525,593,672,756,849,966,1080, %U A298676 1207,1354,1530,1718,1925,2135,2377,2667,2997,3351,3736,4141,4598,5125 %N A298676 Number of partitions of n that can be uniquely recovered from their P-graphs. %C A298676 a(n) is the number of partitions of n that can be uniquely recovered from its P-graph, the simple graph whose vertices are the parts of the partition, two of which are joined by an edge if, and only if, they have a common factor greater than 1. %H A298676 Bernardo Recamán Santos, A unique partition of 200 into 6 parts, Puzzling Stack Exchange, Dec 17 2017. %e A298676 a(1) = 1 because the sole partition of 1 can be recovered from its P-graph, a single vertex. %e A298676 a(2) = 2 because both partitions of 2 can be recovered from their corresponding P-graphs. %t A298676 pgraph[p_] := With[{v = Range[Length[p]]}, Graph[v, UndirectedEdge @@@ Select[Subsets[v, {2}], !CoprimeQ @@ p[[#]] &]]]; %t A298676 a[n_] := Count[Length /@ Gather[pgraph /@ IntegerPartitions[n], IsomorphicGraphQ], 1]; %t A298676 Array[a, 20] %t A298676 (* _Andrey Zabolotskiy_, Jan 30 2018 *) %K A298676 nonn,new %O A298676 1,2 %A A298676 _Bernardo Recamán_, Jan 28 2018 %E A298676 a(23)-a(50) from _Freddy Barrera_, Jan 29 2018 %E A298676 a(51)-a(55) from _Andrey Zabolotskiy_, Jan 30 2018 %I A298682 %S A298682 1,2,4,8,28,92,352,1280,4828,17900,67024,249680,932716,3479132, %T A298682 12987904,48464288,180885628,675045452,2519361712,9402270320, %U A298682 35089981708,130957132220,488739595744,1823999153600,6807261212956,25405037309612,94812904802704 %N A298682 Start with the triangle with 4 markings of the Shield tiling and recursively apply the substitution rule. a(n) is the number of triangles with 4 markings after n iterations. %C A298682 The following substitution rules apply to the tiles: %C A298682 triangle with 6 markings -> 1 hexagon %C A298682 triangle with 4 markings -> 1 square, 2 triangles with 4 markings %C A298682 square -> 1 square, 4 triangles with 6 markings %C A298682 hexagon -> 7 triangles with 6 markings, 3 triangles with 4 markings, 3 squares %C A298682 a(n) is also one more than the number of squares after n iterations when starting with the triangle with 4 markings. %H A298682 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298682 F. Gähler, Matching rules for quasicrystals: the composition-decomposition method, Journal of Non-Crystalline Solids, 153-154 (1993), 160-164. %H A298682 Tilings Encyclopedia, Shield %H A298682 Index entries for linear recurrences with constant coefficients, signature (3,5,-9,2). %F A298682 From _Colin Barker_, Jan 25 2018: (Start) %F A298682 G.f.: (1 + x)*(1 - 2*x - 5*x^2) / ((1 - x)*(1 + 2*x)*(1 - 4*x + x^2)). %F A298682 a(n) = (1/13)*(26 + (-2)^n + (2+sqrt(3))^n*(-7+5*sqrt(3)) - (2-sqrt(3))^n*(7+5*sqrt(3))). %F A298682 a(n) = 3*a(n-1) + 5*a(n-2) - 9*a(n-3) + 2*a(n-4) for n>3. %F A298682 (End) %o A298682 (PARI) /* The function substitute() takes as argument a 4-element vector, where the first, second, third and fourth elements respectively are the number of triangles with 6 markings, the number of triangles with 4 markings, the number of squares and the number of hexagons that are to be substituted. The function returns a vector w, where the first, second, third and fourth elements respectively are the number of triangles with 6 markings, the number of triangles with 4 markings, the number of squares and the number of hexagons resulting from the substitution. */ %o A298682 substitute(v) = my(w=vector(4)); for(k=1, #v, while(v[1] > 0, w[4]++; v[1]--); while(v[2] > 0, w[3]++; w[2]=w[2]+2; v[2]--); while(v[3] > 0, w[3]++; w[1]=w[1]+4; v[3]--); while(v[4] > 0, w[1]=w[1]+7; w[2]=w[2]+3; w[3]=w[3]+3; v[4]--)); w %o A298682 terms(n) = my(v=[0, 1, 0, 0], i=0); while(1, print1(v[2], ", "); i++; if(i==n, break, v=substitute(v))) %o A298682 (PARI) Vec((1 + x)*(1 - 2*x - 5*x^2) / ((1 - x)*(1 + 2*x)*(1 - 4*x + x^2)) + O(x^40)) \\ _Colin Barker_, Jan 25 2018 %Y A298682 Cf. A298678, A298679, A298680, A298681, A298683. %K A298682 nonn,easy,new %O A298682 0,2 %A A298682 _Felix Fröhlich_, Jan 24 2018 %E A298682 More terms from _Colin Barker_, Jan 25 2018 %I A298681 %S A298681 0,4,4,32,80,372,1236,4912,17728,67364,248996,934080,3476400,12993364, %T A298681 48453364,180907472,675001760,2519449092,9402095556,35090331232, %U A298681 130956433168,488740993844,1823996357396,6807266805360,25405026124800,94812927172324,353846503607524 %N A298681 Start with the square tile of the Shield tiling and recursively apply the substitution rule. a(n) is the number of triangles with 6 markings after n iterations. %C A298681 The following substitution rules apply to the tiles: %C A298681 triangle with 6 markings -> 1 hexagon %C A298681 triangle with 4 markings -> 1 square, 2 triangles with 4 markings %C A298681 square -> 1 square, 4 triangles with 6 markings %C A298681 hexagon -> 7 triangles with 6 markings, 3 triangles with 4 markings, 3 squares %H A298681 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298681 F. Gähler, Matching rules for quasicrystals: the composition-decomposition method, Journal of Non-Crystalline Solids, 153-154 (1993), 160-164. %H A298681 Tilings Encyclopedia, Shield %H A298681 Index entries for linear recurrences with constant coefficients, signature (3,5,-9,2). %F A298681 From _Colin Barker_, Jan 25 2018: (Start) %F A298681 G.f.: 4*x*(1 - 2*x) / ((1 - x)*(1 + 2*x)*(1 - 4*x + x^2)). %F A298681 a(n) = (1/39)*(26 + (-1)^(1+n)*2^(5+n) + (3-9*sqrt(3))*(2-sqrt(3))^n + (2+sqrt(3))^n*(3+9*sqrt(3))). %F A298681 a(n) = 3*a(n-1) + 5*a(n-2) - 9*a(n-3) + 2*a(n-4) for n>3. %F A298681 (End) %o A298681 (PARI) /* The function substitute() takes as argument a 4-element vector, where the first, second, third and fourth elements respectively are the number of triangles with 6 markings, the number of triangles with 4 markings, the number of squares and the number of hexagons that are to be substituted. The function returns a vector w, where the first, second, third and fourth elements respectively are the number of triangles with 6 markings, the number of triangles with 4 markings, the number of squares and the number of hexagons resulting from the substitution. */ %o A298681 substitute(v) = my(w=vector(4)); for(k=1, #v, while(v[1] > 0, w[4]++; v[1]--); while(v[2] > 0, w[3]++; w[2]=w[2]+2; v[2]--); while(v[3] > 0, w[3]++; w[1]=w[1]+4; v[3]--); while(v[4] > 0, w[1]=w[1]+7; w[2]=w[2]+3; w[3]=w[3]+3; v[4]--)); w %o A298681 terms(n) = my(v=[0, 0, 1, 0], i=0); while(1, print1(v[1], ", "); i++; if(i==n, break, v=substitute(v))) %o A298681 (PARI) concat(0, Vec(4*x*(1 - 2*x) / ((1 - x)*(1 + 2*x)*(1 - 4*x + x^2)) + O(x^40))) \\ _Colin Barker_, Jan 25 2018 %Y A298681 Cf. A298678, A298679, A298680, A298682, A298683. %K A298681 nonn,easy,new %O A298681 0,2 %A A298681 _Felix Fröhlich_, Jan 24 2018 %E A298681 More terms from _Colin Barker_, Jan 25 2018 %I A298680 %S A298680 0,0,4,12,56,192,756,2748,10408,38544,144452,537900,2009880,7496160, %T A298680 27985684,104424732,389756936,1454515632,5428480356,20259056268, %U A298680 75608443768,282173320704,1053087635252,3930171627900,14667610061160,54740246247120,204293419666564 %N A298680 Start with the triangle with 4 markings of the Shield tiling and recursively apply the substitution rule. a(n) is the number of triangles with 6 markings after n iterations. %C A298680 The following substitution rules apply to the tiles: %C A298680 triangle with 6 markings -> 1 hexagon %C A298680 triangle with 4 markings -> 1 square, 2 triangles with 4 markings %C A298680 square -> 1 square, 4 triangles with 6 markings %C A298680 hexagon -> 7 triangles with 6 markings, 3 triangles with 4 markings, 3 squares %C A298680 a(n) is also the number of hexagonal tiles after n+1 iterations when starting with the triangle with 4 markings. %H A298680 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298680 F. Gähler, Matching rules for quasicrystals: the composition-decomposition method, Journal of Non-Crystalline Solids, 153-154 (1993), 160-164. %H A298680 Tilings Encyclopedia, Shield %H A298680 Index entries for linear recurrences with constant coefficients, signature (3,5,-9,2). %F A298680 From _Colin Barker_, Jan 25 2018: (Start) %F A298680 G.f.: 4*x^2 / ((1 - x)*(1 + 2*x)*(1 - 4*x + x^2)). %F A298680 a(n) = (1/39)*(-26 + (-1)^n*2^(3+n) - (2-sqrt(3))^n*(-9+sqrt(3)) + (2+sqrt(3))^n*(9+sqrt(3))). %F A298680 a(n) = 3*a(n-1) + 5*a(n-2) - 9*a(n-3) + 2*a(n-4) for n>3. %F A298680 (End) %o A298680 (PARI) /* The function substitute() takes as argument a 4-element vector, where the first, second, third and fourth elements respectively are the number of triangles with 6 markings, the number of triangles with 4 markings, the number of squares and the number of hexagons that are to be substituted. The function returns a vector w, where the first, second, third and fourth elements respectively are the number of triangles with 6 markings, the number of triangles with 4 markings, the number of squares and the number of hexagons resulting from the substitution. */ %o A298680 substitute(v) = my(w=vector(4)); for(k=1, #v, while(v[1] > 0, w[4]++; v[1]--); while(v[2] > 0, w[3]++; w[2]=w[2]+2; v[2]--); while(v[3] > 0, w[3]++; w[1]=w[1]+4; v[3]--); while(v[4] > 0, w[1]=w[1]+7; w[2]=w[2]+3; w[3]=w[3]+3; v[4]--)); w %o A298680 terms(n) = my(v=[0, 1, 0, 0], i=0); while(1, print1(v[1], ", "); i++; if(i==n, break, v=substitute(v))) %o A298680 (PARI) concat(vector(2), Vec(4*x^2 / ((1 - x)*(1 + 2*x)*(1 - 4*x + x^2)) + O(x^40))) \\ _Colin Barker_, Jan 25 2018 %Y A298680 Cf. A298678, A298679, A298681, A298682, A298683. %K A298680 nonn,easy,new %O A298680 0,3 %A A298680 _Felix Fröhlich_, Jan 24 2018 %E A298680 More terms from _Colin Barker_, Jan 25 2018 %I A298679 %S A298679 0,3,6,33,102,423,1494,5745,21102,79431,295086,1103985,4114710, %T A298679 15367143,57329286,213999153,798569022,2980473543,11122931934, %U A298679 41512040625,154923657702,578185735911,2157812994486,8053078824945,30054477139470,112164880064583 %N A298679 Start with the hexagonal tile of the Shield tiling and recursively apply the substitution rule. a(n) is the number of square tiles after n iterations. %C A298679 The following substitution rules apply to the tiles: %C A298679 triangle with 6 markings -> 1 hexagon %C A298679 triangle with 4 markings -> 1 square, 2 triangles with 4 markings %C A298679 square -> 1 square, 4 triangles with 6 markings %C A298679 hexagon -> 7 triangles with 6 markings, 3 triangles with 4 markings, 3 squares %C A298679 a(n) is also the number of triangles with 4 markings after n+1 iterations when starting with the hexagonal tile. %C A298679 a(n) is also the number of square tiles after n+1 iterations when starting with the hexagonal tile. %H A298679 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298679 F. Gähler, Matching rules for quasicrystals: the composition-decomposition method, Journal of Non-Crystalline Solids, 153-154 (1993), 160-164. %H A298679 Tilings Encyclopedia, Shield %H A298679 Index entries for linear recurrences with constant coefficients, signature (2,7,-2). %F A298679 From _Colin Barker_, Jan 25 2018: (Start) %F A298679 G.f.: 3*x / ((1 + 2*x)*(1 - 4*x + x^2)). %F A298679 a(n) = (1/26)*(-3*(-1)^n*2^(2+n) + (6-5*sqrt(3))*(2-sqrt(3))^n + (2+sqrt(3))^n*(6+5*sqrt(3))). %F A298679 a(n) = 2*a(n-1) + 7*a(n-2) - 2*a(n-3) for n>2. %F A298679 (End) %o A298679 (PARI) /* The function substitute() takes as argument a 4-element vector, where the first, second, third and fourth elements respectively are the number of triangles with 6 markings, the number of triangles with 4 markings, the number of squares and the number of hexagons that are to be substituted. The function returns a vector w, where the first, second, third and fourth elements respectively are the number of triangles with 6 markings, the number of triangles with 4 markings, the number of squares and the number of hexagons resulting from the substitution. */ %o A298679 substitute(v) = my(w=vector(4)); for(k=1, #v, while(v[1] > 0, w[4]++; v[1]--); while(v[2] > 0, w[3]++; w[2]=w[2]+2; v[2]--); while(v[3] > 0, w[3]++; w[1]=w[1]+4; v[3]--); while(v[4] > 0, w[1]=w[1]+7; w[2]=w[2]+3; w[3]=w[3]+3; v[4]--)); w %o A298679 terms(n) = my(v=[0, 0, 0, 1], i=0); while(1, print1(v[3], ", "); i++; if(i==n, break, v=substitute(v))) %o A298679 (PARI) concat(0, Vec(3*x / ((1 + 2*x)*(1 - 4*x + x^2)) + O(x^40))) \\ _Colin Barker_, Jan 25 2018 %Y A298679 Cf. A298678, A298680, A298681, A298682, A298683. %K A298679 nonn,easy,new %O A298679 0,2 %A A298679 _Felix Fröhlich_, Jan 24 2018 %E A298679 More terms from _Colin Barker_, Jan 25 2018 %I A298678 %S A298678 1,0,7,12,73,216,919,3204,12409,45408,171271,635580,2379241,8865000, %T A298678 33113527,123523572,461111833,1720661616,6422058919,23966525484, %U A298678 89446140169,333813840888,1245817611991,4649439829860,17351975261881,64758394108800,241681735391047 %N A298678 Start with the hexagonal tile of the Shield tiling and recursively apply the substitution rule. a(n) is the number of hexagonal tiles after n iterations. %C A298678 The following substitution rules apply to the tiles: %C A298678 triangle with 6 markings -> 1 hexagon %C A298678 triangle with 4 markings -> 1 square, 2 triangles with 4 markings %C A298678 square -> 1 square, 4 triangles with 6 markings %C A298678 hexagon -> 7 triangles with 6 markings, 3 triangles with 4 markings, 3 squares %C A298678 For n > 0, a(n) is also the number of triangles with 6 markings after n iterations when starting with the hexagon. %C A298678 a(n) is also the number of triangles with 6 markings after n iterations when starting with the triangle with 6 markings. %C A298678 a(n) is also the number of hexagons after n iterations when starting with the triangle with 6 markings. %H A298678 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298678 F. Gähler, Matching rules for quasicrystals: the composition-decomposition method, Journal of Non-Crystalline Solids, 153-154 (1993), 160-164. %H A298678 Tilings Encyclopedia, Shield %H A298678 Index entries for linear recurrences with constant coefficients, signature (2,7,-2). %F A298678 G.f.: (1-2*x)/((1+2*x)*(1-4*x+x^2)). - _Joerg Arndt_, Jan 25 2018 %F A298678 13*a(n) = A077235(n) + 8*(-2)^n. - _Bruno Berselli_, Jan 25 2018 %F A298678 From _Colin Barker_, Jan 25 2018: (Start) %F A298678 a(n) = (1/26)*((-1)^n*2^(4+n) + (5-2*sqrt(3))*(2-sqrt(3))^n + (2+sqrt(3))^n*(5+2*sqrt(3))). %F A298678 a(n) = 2*a(n-1) + 7*a(n-2) - 2*a(n-3) for n>2. %F A298678 (End) %o A298678 (PARI) /* The function substitute() takes as argument a 4-element vector, where the first, second, third and fourth elements respectively are the number of triangles with 6 markings, the number of triangles with 4 markings, the number of squares and the number of hexagons that are to be substituted. The function returns a vector w, where the first, second, third and fourth elements respectively are the number of triangles with 6 markings, the number of triangles with 4 markings, the number of squares and the number of hexagons resulting from the substitution. */ %o A298678 substitute(v) = my(w=vector(4)); for(k=1, #v, while(v[1] > 0, w[4]++; v[1]--); while(v[2] > 0, w[3]++; w[2]=w[2]+2; v[2]--); while(v[3] > 0, w[3]++; w[1]=w[1]+4; v[3]--); while(v[4] > 0, w[1]=w[1]+7; w[2]=w[2]+3; w[3]=w[3]+3; v[4]--)); w %o A298678 terms(n) = my(v=[0, 0, 0, 1], i=0); while(1, print1(v[4], ", "); i++; if(i==n, break, v=substitute(v))) %o A298678 (PARI) Vec((1-2*x)/((1+2*x)*(1-4*x+x^2)) + O(x^40)) \\ _Colin Barker_, Jan 25 2018 %Y A298678 Cf. A298679, A298680, A298681, A298682, A298683. %K A298678 nonn,easy,new %O A298678 0,3 %A A298678 _Felix Fröhlich_, Jan 24 2018 %E A298678 More terms from _Colin Barker_, Jan 25 2018 %I A296205 %S A296205 1,6,10,12,14,15,18,20,21,22,26,28,33,34,35,36,38,39,44,45,46,50,51, %T A296205 52,55,57,58,62,63,65,68,69,74,75,76,77,82,85,86,87,91,92,93,94,95,98, %U A296205 99,100,106,111,115,116,117,118,119,122,123,124,129,133,134,141,142,143,145,146,147,148,153,155,158,159,161 %N A296205 Numbers n such that Product_{d|n^2, gcd(d,n^2/d) is prime} gcd(d,n^2/d) = n^2. %F A296205 a(n) = A000196(A296204(n)). %Y A296205 Cf. A000196, A295666, A296204. %Y A296205 Cf. A006881, A054753, A085986 (seem to be subsequences). %K A296205 nonn,new %O A296205 1,2 %A A296205 _Antti Karttunen_, Dec 18 2017 %I A296204 %S A296204 1,36,100,144,196,225,324,400,441,484,676,784,1089,1156,1225,1296, %T A296204 1444,1521,1936,2025,2116,2500,2601,2704,3025,3249,3364,3844,3969, %U A296204 4225,4624,4761,5476,5625,5776,5929,6724,7225,7396,7569,8281,8464,8649,8836,9025,9604,9801,10000,11236,12321,13225,13456,13689,13924,14161 %N A296204 Numbers n such that Product_{d|n, gcd(d,n/d) is prime} gcd(d,n/d) = n; the fixed points of A295666. %Y A296204 Cf. A295666, A296205 (the square roots). %K A296204 nonn,new %O A296204 1,2 %A A296204 _Antti Karttunen_, Dec 18 2017 %I A298912 %S A298912 1,2,9,21,25,38,45,57,93,105,121,165,194,201,202,205,206,218,253,261, %T A298912 301,315,325,326,357,361,381,385,422,453,477,482,494,506,538,542,554, %U A298912 603,614,626,633,662,746,758,765,801,841,861,873,897,921,925,934,1005,1017 %N A298912 Numbers n such that the number of groups of order n equals the number of groups of order n + 1. %H A298912 H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644. %H A298912 Gordon Royle, Numbers of Small Groups %H A298912 Index entries for sequences related to groups %F A298912 Sequence is { n | A000001(n+1) = A000001(n) }. %p A298912 with(GroupTheory): %p A298912 for n from 1 to 300 do if NumGroups(n+1) = NumGroups(n) then print(n); fi; od; %o A298912 (GAP) Filtered([1..500], n -> NumberSmallGroups(n) = NumberSmallGroups(n+1)); %Y A298912 Cf. A000001. Numbers n having precisely k groups of orders n and n + 1: A295230 (k=2), A295990 (k=4), A295991 (k=5), A295992 (k=6), A295993 (k=8), A298427 (k=9), A (k=10), A295994 (k=11), A295995 (k=15). %K A298912 nonn,new %O A298912 1,2 %A A298912 _Muniru A Asiru_, Jan 28 2018 %I A298911 %S A298911 820,1220,1530,2020,2070,2610,2756,3366,3620,4230,4550,4770,4820,5310, %T A298911 5620,5742,5950,6370,6650,7038,7470,8010,8020,8050,8118,8164,8330, %U A298911 8420,8874,9220,9306,9310,9316,9630,10170,10420,10494,10820,11050 %N A298911 Numbers n such that there are precisely 20 groups of order n. %H A298911 H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644. %H A298911 Gordon Royle, Numbers of Small Groups %H A298911 Index entries for sequences related to groups %F A298911 Sequence is { n | A000001(n) = 20 }. %e A298911 For n = 820, the 20 groups are (C41 : C5) : C4, C4 x (C41 : C5), C41 x (C5 : C4), C5 x (C41 : C4), C205 : C4, C820, (C41 : C5) : C4, C2 x ((C41 : C5) : C2), C2 x C2 x (C41 : C5), C5 x (C41 : C4), C41 x (C5 : C4), C205 : C4, C205 : C4, C205 : C4, C205 : C4, D10 x D82, C10 x D82, C82 x D10, D820, C410 x C2 where C, D mean the Cyclic, Dihedral groups of the stated order and the symbols x and : mean direct and semidirect products respectively. %p A298911 with(GroupTheory): %p A298911 for n from 1 to 10^4 do if NumGroups(n) = 20 then print(n); fi; od; %Y A298911 Cf. A000001. Cyclic numbers A003277. Numbers n such that there are precisely k groups of order n: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A249155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), this sequence (k=20). %K A298911 nonn,new %O A298911 1,1 %A A298911 _Muniru A Asiru_, Jan 28 2018 %I A298910 %S A298910 1029,5145,6591,7803,8001,11319,11739,12789,17157,17493,20577,21567, %T A298910 23667,23877,27993,31311,32955,33411,34671,34713,39015,39753,40005, %U A298910 42189,42861,45675,47691,48363,49833,50673,55083,55629,57603,58539,58695,60501 %N A298910 Numbers n such that there are precisely 19 groups of order n. %H A298910 H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644. %H A298910 Gordon Royle, Numbers of Small Groups %H A298910 Index entries for sequences related to groups %F A298910 Sequence is { n | A000001(n) = 19 }. %e A298910 For n = 1029, the 19 groups are C1029, C147 x C7, C3 x ((C7 x C7) : C7), C3 x (C49 : C7), C21 x C7 x C7, C343 : C3, C49 x (C7 : C3), C7 x (C49 : C3), (C49 x C7) : C3, (C49 x C7) : C3, ((C7 x C7) : C7) : C3, ((C7 x C7) : C7) : C3, ((C7 x C7) : C7) : C3, (C49 : C7) : C3, C7 x ((C7 x C7) : C3), C7 x ((C7 x C7) : C3), (C7 x C7 x C7) : C3, (C7 x C7 x C7) : C3, C7 x C7 x (C7 : C3) where C means the Cyclic group of the stated order and the symbols x and : mean direct and semidirect products respectively. %p A298910 with(GroupTheory): %p A298910 for n from 1 to 3*10^5 do if NumGroups(n) = 19 then print(n); fi; od; %Y A298910 Cf. A000001. Cyclic numbers A003277. Numbers n such that there are precisely k groups of order n: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A249155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), this sequence (k=19), A298911 (k=20). %K A298910 nonn,new %O A298910 1,1 %A A298910 _Muniru A Asiru_, Jan 28 2018 %I A298909 %S A298909 156,342,444,666,732,876,930,1164,1308,1314,1830,1884,1962,2172,2286, %T A298909 2316,2748,2892,2934,3258,3324,3582,3675,3756,4044,4125,4188,4422, %U A298909 4476,4530,4764,4878,4908,4970,5050,5052,5196,5430,5445,5481,5484,5526,6330,6492,6822,6924 %N A298909 Numbers n such that there are precisely 18 groups of order n. %H A298909 H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644. %H A298909 Gordon Royle, Numbers of Small Groups %H A298909 Index entries for sequences related to groups %F A298909 Sequence is { n | A000001(n) = 18 }. %e A298909 For n = 156, the 18 groups are (C13 : C4) : C3, C4 x (C13 : C3), C13 x (C3 : C4), C3 x (C13 : C4), C39 : C4, C156, (C13 : C4) : C3, C2 x ((C13 : C3) : C2), C3 x (C13 : C4), C39 : C4, S3 x D26, C2 x C2 x (C13 : C3), C13 x A4, (C26 x C2) : C3, C6 x D26, C26 x S3, D156, C78 x C2 where C, D mean Cyclic, Dihedral groups of the stated order and S, A mean the Symmetric, Alternating groups of the stated degree. The symbols x and : mean direct and semidirect products respectively. %p A298909 with(GroupTheory): %p A298909 for n from 1 to 10^4 do if NumGroups(n) = 18 then print(n); fi; od; %o A298909 (GAP) Filtered([1..2015], n -> NumberSmallGroups(n) = 18); %Y A298909 Cf. A000001. Cyclic numbers A003277. Numbers n such that there are precisely k groups of order n: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A249155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), this sequence (k=18), A298910 (k=19), A298911 (k=20). %K A298909 nonn,new %O A298909 1,1 %A A298909 _Muniru A Asiru_, Jan 28 2018 %I A296604 %S A296604 0,0,0,0,1,2,1,4,2,4,3,4,2,8,1,3,3,4,0,6,1,4,0,2,0,4,3,0,0,1,0,5,0,1, %T A296604 0,0,3,0,0,0,0,7,0,0,0,0,1,0,0,0,0,7,0,0,0,0,0,0,0,0,0,5,0,0,0 %N A296604 Number of Johnson solids with n faces. %C A296604 Sum(n>0, a(n)) = 92, the number of Johnson solids, as conjectured by Johnson and proved by Zalgaller. %C A296604 a(n) > 0 if and only if n is a member of A296604. %H A296604 Norman W. Johnson, Convex Polyhedra with Regular Faces, Canadian Journal of Mathematics, 18 (1966), 169-200. %H A296604 Eric W. Weisstein, MathWorld: Johnson Solid %H A296604 Wikipedia, List of Johnson solids %H A296604 Victor A. Zalgaller, Convex Polyhedra with Regular Faces, Zap. Nauchn. Sem. LOMI, 1967, Volume 2. Pages 5-221 (Mi znsl1408). %F A296604 a(62) = 5. %F A296604 a(n) = 0 for n > 62. %e A296604 The square pyramid is the only Johnson solid with five faces, so a(5) = 1. %Y A296604 Cf. A181708, A242731, A296602, A296603. %K A296604 nonn,new %O A296604 1,6 %A A296604 _Jonathan Sondow_, Jan 28 2018 %I A296603 %S A296603 5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,21,22,24,26,27,30,32,34,37, %T A296603 42,47,52,62 %N A296603 Number of faces a Johnson solid can have. %C A296603 Distinct terms in A242731, sorted. %C A296603 n is a member if and only if A296604(n) > 0. %H A296603 Norman W. Johnson, Convex Polyhedra with Regular Faces, Canadian Journal of Mathematics, 18 (1966), 169-200. %H A296603 Eric W. Weisstein, MathWorld: Johnson Solid %H A296603 Wikipedia, List of Johnson solids %H A296603 Victor A. Zalgaller, Convex Polyhedra with Regular Faces, Zap. Nauchn. Sem. LOMI, 1967, Volume 2. Pages 5-221 (Mi znsl1408). %e A296603 The square pyramid is the Johnson solid with the fewest faces, namely, 5, so a(1) = 5. %Y A296603 Cf. A181708, A242731, A296602, A296604. %K A296603 nonn,fini,full,new %O A296603 1,1 %A A296603 _Jonathan Sondow_, Jan 28 2018 %I A298008 %S A298008 4,14,22,32,43,52,62,73,82,91,104,111,121,133,141,152,162,172,181,194, %T A298008 200,211,223,232,241,252,262,272,282,291,301,313,320,332,342,352,361, %U A298008 372,382,391,402,411,421,433,442,451,463,471,481,492,502,510,522,530,542,551,562,572,581,592,602,613,620,631,643 %N A298008 a(n) = f(n-1,n) + 10*(n-1), where f(a,b) is the number of primes in the range [10*a,10*b]. %F A298008 a(n) = A038800(n-1) + 10*(n-1). - _Michel Marcus_, Jan 11 2018 %e A298008 The first term has the number of prime numbers between 0 and 9: 4. Since the numbers in this first range are smaller than 10, the left digit would be a zero (not represented). The second term has the number of prime numbers between 10 and 19 (4) and since it was counted in the range between 10 and 19 it represents this range with the one in the first digit in the left: 14. The third element is 22 as there are 2 primes between 20 and 29. And so on. Larger element: there is only one prime between 120 and 129, hence a(13)=121. %t A298008 Block[{p = 1, k}, k = 10^p; Array[Apply[Subtract, PrimePi[{k #, k (# - 1)}]] + (# - 1) k &, 67]] (* _Michael De Vlieger_, Jan 11 2018 *) %o A298008 (Python) %o A298008 # Generates all elements of the sequence smaller than 'last' %o A298008 last = 1000 %o A298008 p=[2] %o A298008 c=1 %o A298008 for i in range(3,last+2,2): %o A298008 prime = True %o A298008 for j in p: %o A298008 if i%j == 0: %o A298008 prime=False; %o A298008 break; %o A298008 if prime == True: %o A298008 p.append(i) %o A298008 c = c + 1 %o A298008 ii = int(i/10)*10 %o A298008 if i-ii == 1: %o A298008 if prime == True: %o A298008 print '%d,' % (ii-10+c-1), %o A298008 c = 1 %o A298008 else: %o A298008 print '%d,' % (ii-10+c), %o A298008 c = 0 %Y A298008 Cf. A038800. %K A298008 nonn,new %O A298008 1,1 %A A298008 _Luis F.B.A. Alexandre_, Jan 10 2018 %E A298008 Edited by _N. J. A. Sloane_, Jan 28 2018 %I A295076 %S A295076 6,10,12,14,20,22,24,26,28,30,34,38,40,42,44,46,48,52,54,56,58,60,62, %T A295076 66,68,70,74,76,78,80,82,84,86,88,90,92,94,96,102,104,106,108,110,112, %U A295076 114,116,118,120,122,124,126,130,132,134,136,138,140,142,146,148 %N A295076 Numbers n > 1 such that n and sigma(n) have the same smallest prime factor. %C A295076 Supersequence of A088829; this sequence contains also odd numbers: 441, 1521, 3249, 3969, 8649, 11025, ... %C A295076 Even terms of A000396 (perfect numbers) are a subsequence. %C A295076 Subsequence of A295078. %C A295076 Numbers n such that A020639(n) = A020639(sigma(n)). %C A295076 Numbers n such that A020639(n) = A071189(n). %e A295076 30 = 2*3*5 and sigma(30) = 72 = 2^3*3^2 hence 30 is in the sequence. %p A295076 select(t -> min(numtheory:-factorset(t))=min(numtheory:-factorset(numtheory:-sigma(t))), [$2..1000]); # _Robert Israel_, Nov 14 2017 %t A295076 Rest@ Select[Range@ 150, SameQ @@ Map[FactorInteger[#][[1, 1]] &, {#, DivisorSigma[1, #]}] &] (* _Michael De Vlieger_, Nov 13 2017 *) %o A295076 (MAGMA) [n: n in [2..1000000] | Minimum(PrimeDivisors(SumOfDivisors(n))) eq Minimum(PrimeDivisors(n))] %o A295076 (PARI) isok(n) = factor(n)[1,1] == factor(sigma(n))[1,1]; \\ _Michel Marcus_, Nov 14 2017 %Y A295076 Cf. A000203, A020639, A071189, A088829, A295078. %Y A295076 Cf. A071834 (numbers n such that n and sigma(n) have the same largest prime factor). %K A295076 nonn,new %O A295076 1,1 %A A295076 _Jaroslav Krizek_, Nov 13 2017 %E A295076 Added n>1 to definition - _N. J. A. Sloane_, Feb 03 2018 %I A295043 %S A295043 1,0,3,7,0,31,0,127,217,381,889,0,3937,8191,11811,27559,57337,131071, %T A295043 253921,524287,1040257,1777447,4063201,7281799,16646017,32247967, %U A295043 66584449,116522119,225735769,516026527,1073602561,2147483647,4294434817,7515217927,15032385529 %N A295043 a(n) is the largest number k such that sigma(k) = 2^n or 0 if no such k exists. %C A295043 If a(n) > 0, then it is a term of A046528 (numbers that are a product of distinct Mersenne primes). %F A295043 a(A078426(n)) = 0. %F A295043 a(A180221(n)) > 0. %F A295043 a(n) <= 2^n - 1 with equality when n is a Mersenne exponent (A000043). - _Michael B. Porter_, Nov 14 2017 %e A295043 a(0) = 1 because 1 is the largest number k with sigma(k) = 1 = 2^0. %e A295043 a(5) = 31 because 31 is the largest number k with sigma(k) = 32 = 2^5. %e A295043 a(6) = 0 because there is no number k with sigma(k) = 64 = 2^6. %o A295043 (PARI) a(n) = {local(r, k); r=0; for(k=1, 2^n, if(sigma(k) == 2^n, r=k)); return(r)}; \\ _Michael B. Porter_, Nov 14 2017 %o A295043 (PARI) a(n) = forstep(k=2^n, 1, -1, if (sigma(k)==2^n, return (k))); return (0) \\ _Rémy Sigrist_, Jan 08 2018 %Y A295043 Cf. A000043, A000203, A046528, A048947, A057637, A180221. %Y A295043 Cf. A247956 (the smallest number k instead of the largest). %Y A295043 Cf. A078426 (no solution to the equation sigma(x)=2^n). %Y A295043 A000668 (Mersenne primes) is a subsequence. %K A295043 nonn,new %O A295043 0,3 %A A295043 _Jaroslav Krizek_, Nov 13 2017 %I A297996 %S A297996 2,3,5,2,6,3,7,4,8,5,9,6,10,7,11,8,12,9,13,10,14,11,15,12,16,13,17,14, %T A297996 18,15,19,16,20,17,21,18,22,19,23,20,24,21,25,22,26,23,27,24,28,25,29, %U A297996 26,30,27,31,28,32,29,33,30,34,31,35,32,36,33,37,34,38,35 %N A297996 a(1)=2, a(2)=3, a(3)=5 and a(n) = (a(1) + a(2) + a(3) + ... + a(n-1))/a(n-1). %H A297996 Colin Barker, Table of n, a(n) for n = 1..1000 %H A297996 Index entries for linear recurrences with constant coefficients, signature (1,1,-1). %F A297996 a(n) = A168230(n+1) for n >= 3. %F A297996 From _Colin Barker_, Jan 29 2018: (Start) %F A297996 G.f.: x*(2 + x - 4*x^3 + 2*x^4) / ((1 - x)^2*(1 + x)). %F A297996 a(n) = n/2 for n>2 and even. %F A297996 a(n) = (n+7)/2 for n>2 and odd. %F A297996 a(n) = a(n-1) + a(n-2) - a(n-3) for n>5. %F A297996 (End) %t A297996 Nest[Append[#, Total[#]/Last[#]] &, Prime@ Range@ 3, 67] (* _Michael De Vlieger_, Jan 10 2018 *) %o A297996 (PARI) lista(nn) = {va = vector(nn); for (n=1, 3, va[n] = prime(n)); for (n=4, nn, va[n] = sum(k=1, n-1, va[k])/va[n-1];); va;} \\ _Michel Marcus_, Jan 10 2018 %o A297996 (PARI) Vec(x*(2 + x - 4*x^3 + 2*x^4) / ((1 - x)^2*(1 + x)) + O(x^100)) \\ _Colin Barker_, Jan 29 2018 %Y A297996 Cf. A168230. %K A297996 nonn,easy,new %O A297996 1,1 %A A297996 _Mateusz Pasternak_, Jan 10 2018 %E A297996 More terms from _Michel Marcus_, Jan 10 2018 %I A297626 %S A297626 1,2,2,1,3,1,3,1,1,3,2,1,4,2,1,4,2,1,1,5,2,1,1,5,3,1,1,5,3,2,1,5,3,2, %T A297626 1,1,6,3,2,1,1,6,3,2,1,1,1,6,4,2,1,1,1,6,4,2,2,1,1,6,4,3,2,1,1,7,4,3, %U A297626 2,1,1,7,4,3,2,1,1,1,7,5,3,2,1,1,1,8,5,3,2,1,1,1,8,5,3,2,2,1,1,8,5,3,2,2,1,1,1,9,5,3,2,2,1,1,1,8,5,4,3,2,1,1,1,9,5,4,3,2,1,1,1,9,6,4,3,2,1,1,1,9,6,4,3,2,1,1,1,1,9,6,4,3,2,2,1,1,1,10,6,4,3,2,2,1,1,1 %N A297626 Triangle read by rows in which row n gives a partition of n with the most subpartitions. %C A297626 A partition and its conjugate have the same number of subpartitions; in the case of ties, we take the lexicographically earliest partition. %H A297626 Mathoverflow, Upper bound for number of subpartitions of a partition, posted Sep 16 2017 %e A297626 Triangle begins %e A297626 1 %e A297626 2 %e A297626 2 1 %e A297626 3 1 %e A297626 3 1 1 %e A297626 3 2 1 %e A297626 4 2 1 %e A297626 4 2 1 1 %e A297626 5 2 1 1 %e A297626 5 3 1 1 %e A297626 5 3 2 1 %e A297626 5 3 2 1 1 %e A297626 6 3 2 1 1 %e A297626 6 3 2 1 1 1 %e A297626 6 4 2 1 1 1 %e A297626 6 4 2 2 1 1 %e A297626 6 4 3 2 1 1 %e A297626 7 4 3 2 1 1 %e A297626 7 4 3 2 1 1 1 %e A297626 7 5 3 2 1 1 1 %e A297626 8 5 3 2 1 1 1 %e A297626 8 5 3 2 2 1 1 %e A297626 8 5 3 2 2 1 1 1 %e A297626 9 5 3 2 2 1 1 1 %e A297626 8 5 4 3 2 1 1 1 %e A297626 9 5 4 3 2 1 1 1 %e A297626 9 6 4 3 2 1 1 1 %e A297626 9 6 4 3 2 1 1 1 1 %e A297626 9 6 4 3 2 2 1 1 1 %Y A297626 Cf. A116480, A117500. %K A297626 nonn,tabf,new %O A297626 1,2 %A A297626 _Brian Hopkins_, Jan 04 2018 %I A296056 %S A296056 1,-2,-1400,-679140000,-122489812645200000, %T A296056 -6931927717187904217987200000, %U A296056 -114287375178291587421201860354580633600000,-527655997339226839875614785993553970321322576128000000000,-666218073328701414704702576237379472614149140939534461737723520000000000000 %N A296056 Determinant of the inverse of the matrix A_n, where A_n is the n X n matrix defined by A_n[i,j] = 1/C(i+j-2) for 1 <= i,j <= n, and C(k) is the k-th Catalan number (A000108). %C A296056 It is conjectured that a(n) is an integer for all n. %C A296056 The contributor suggests the name "Catbert matrix" for the matrix A_n, based on its similarity to the Hilbert matrix and its relation to the Catalan numbers. %H A296056 Tom Richardson, Table of n, a(n) for n = 1..29 %H A296056 Tom Richardson, Table of n, a(n) for n = 1..100 %t A296056 f[n_] := Denominator@ Det@ Table[ 1/CatalanNumber[i + j -2], {i, n}, {j, n}]; Array[f, 9] (* _Robert G. Wilson v_, Jan 05 2018 *) %o A296056 (PARI) a(n) = 1/matdet(matrix(n,n,i,j,(i+j-1)/binomial(2*i+2*j-4,i+j-2))) %Y A296056 Cf. A000108, A005249, A062381. %K A296056 sign,new %O A296056 1,2 %A A296056 _Tom Richardson_, Dec 03 2017 %I A298739 %S A298739 0,0,1,-1,1,-1,4,-3,0,-1,4,-4,1,-1,13,-13,4,-4,4,-3,0,-1,14,-13,0,3, %T A298739 -1,-3,3,-3,50,-50,1,-1,13,-13,1,0,12,-13,5,-5,3,-2,0,-1,51,-50,3,-4, %U A298739 4,-4,14,-13,11,-11,0,-1,12,-12,1,2,263,-266,3,-3 %N A298739 First differences of A000001 (the number of groups of order n). %H A298739 H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644. %H A298739 Gordon Royle, Numbers of Small Groups %H A298739 Index entries for sequences related to groups %F A298739 a(n) = A000001(n+1) - A000001(n). %e A298739 There is only one group of order 1 and of order 2, so a(1) = A000001(2) - A000001(1) = 1 - 1 = 0. %e A298739 There are 2 groups of order 4 and 3 is a cyclic number, so a(3) = A000001(4) - A000001(3) = 2 - 1 = 1. %p A298739 with(GroupTheory): seq((NumGroups(n+1) - NumGroups(n), n=1..500)); %o A298739 (GAP) List([1..700],n -> NumberSmallGroups(n+1) - NumberSmallGroups(n)); %Y A298739 Cf. A000001 (Number of groups of order n). %K A298739 sign,new %O A298739 1,7 %A A298739 _Muniru A Asiru_, Jan 25 2018 %I A298437 %S A298437 83132,86049,173529,492830,704241,889406 %N A298437 Numbers n such that there are precisely 16 groups of orders n and n + 1. %C A298437 Equivalently, lower member of consecutive terms of A295161. %H A298437 H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644. %H A298437 Gordon Royle, Numbers of Small Groups %H A298437 Index entries for sequences related to groups %F A298437 Sequence is { n | A000001(n) = 16, A000001(n+1) = 16 }. %e A298437 For n = 83132, A000001(83132) = A000001(83133) = 16. %e A298437 For n = 173529, A000001(173529) = A000001(173530) = 16. %e A298437 For n = 492830, A000001(492830) = A000001(492831) = 16. %p A298437 with(GroupTheory): for n from 1 to 10^6 do if [NumGroups(n), NumGroups(n+1)] = [16, 16] then print(n); fi; od; %Y A298437 Cf. A000001. Subsequence of A295161 (Numbers n having precisely 16 groups of order n). Numbers n having precisely k groups of orders n and n+1: A295230 (k=2), A295990 (k=4), A295991 (k=5), A295992 (k=6), A295993 (k=8), A298427 (k=9), A298428 (k=10), A295994 (k=11), A298429 (k=12), A298430 (k=13), A298431 (k=14), A295995 (k=15), this sequence (k=16). %K A298437 nonn,more,new %O A298437 1,1 %A A298437 _Muniru A Asiru_, Jan 19 2018 %I A298431 %S A298431 4328,22311,29864,57896,75368,99368,120807,130664,131943,152295, %T A298431 157287,164072,180327,184232,212456,236583,268712,276392,331112, %U A298431 338792,381927 %N A298431 Numbers n such that there are precisely 14 groups of orders n and n + 1. %C A298431 Equivalently, lower member of consecutive terms of A294155. %H A298431 H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644. %H A298431 Gordon Royle, Numbers of Small Groups %H A298431 Index entries for sequences related to groups %F A298431 Sequence is { n | A000001(n) = 14, A000001(n+1) = 14 }. %e A298431 For n = 4328, A000001(4328) = A000001(4329) = 14. %e A298431 For n = 22311, A000001(22311) = A000001(22312) = 14. %e A298431 For n = 29864, A000001(29864) = A000001(29865) = 14. %p A298431 with(GroupTheory): for n from 1 to 10^5 do if [NumGroups(n), NumGroups(n+1)] = [14, 14] then print(n); fi; od; %Y A298431 Cf. A000001. Subsequence of A294155 (Numbers n having precisely 14 groups of order n). Numbers n having precisely k groups of orders n and n+1: A295230 (k=2), A295990 (k=4), A295991 (k=5), A295992 (k=6), A295993 (k=8), A298427 (k=9), A298428 (k=10), A295994 (k=11), A298429 (k=12), A298430 (k=13), this sequence (k=14), A295995 (k=15). %K A298431 nonn,more,new %O A298431 1,1 %A A298431 _Muniru A Asiru_, Jan 19 2018 %I A298430 %S A298430 82323,390446,622916,774548,793827,876932 %N A298430 Numbers n such that there are precisely 13 groups of orders n and n + 1. %C A298430 Equivalently, lower member of consecutive terms of A292896. %H A298430 H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644. %H A298430 Gordon Royle, Numbers of Small Groups %H A298430 Index entries for sequences related to groups %F A298430 Sequence is { n | A000001(n) = 13, A000001(n+1) = 13 }. %e A298430 For n = 82323, A000001(82323) = A000001(82324) = 13. %e A298430 For n = 390446, A000001(390446) = A000001(390447) = 13. %e A298430 For n = 622916, A000001(622916) = A000001(622917) = 13. %p A298430 with(GroupTheory): for n from 1 to 10^6 do if [NumGroups(n), NumGroups(n+1)] = [13, 13] then print(n); fi; od; %Y A298430 Cf. A000001. Subsequence of A292896 (Numbers n having precisely 13 groups of order n). Numbers n having precisely k groups of orders n and n+1: A295230 (k=2), A295990 (k=4), A295991 (k=5), A295992 (k=6), A295993 (k=8), A298427 (k=9), A298428 (k=10), A295994 (k=11), A298429 (k=12), this sequence (k=13), A298431 (k=14), A295995 (k=15). %K A298430 nonn,more,new %O A298430 1,1 %A A298430 _Muniru A Asiru_, Jan 19 2018 %I A298429 %S A298429 30135,76312,130890,173445,356610 %N A298429 Numbers n such that there are precisely 12 groups of orders n and n + 1. %C A298429 Equivalently, lower member of consecutive terms of A249555. %H A298429 H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644. %H A298429 Gordon Royle, Numbers of Small Groups %H A298429 Index entries for sequences related to groups %F A298429 Sequence is { n | A000001(n) = 12, A000001(n+1) = 12 }. %e A298429 For n = 30135, A000001(30135) = A000001(30136) = 12. %e A298429 For n = 76312, A000001(76312) = A000001(76313) = 12. %e A298429 For n = 130890, A000001(130890) = A000001(130891) = 12. %p A298429 withGroupTheory): for n from 1 to 10^6 do if [NumGroups(n), NumGroups(n+1)] = [12, 12] then print(n); fi; od; %Y A298429 Cf. A000001. Subsequence of A249555 (Numbers n having precisely 12 groups of order n). Numbers n having precisely k groups of orders n and n+1: A295230 (k=2), A295990 (k=4), A295991 (k=5), A295992 (k=6), A295993 (k=8), A298427 (k=9), A298428 (k=10), A295994 (k=11), this sequence (k=12), A298430 (k=13), A298431 (k=14), A295995 (k=15). %K A298429 nonn,more,new %O A298429 1,1 %A A298429 _Muniru A Asiru_, Jan 19 2018 %I A298428 %S A298428 13914,15974,77234,99126,107205,122675,128894,187473,188265,204134 %N A298428 Numbers n such that there are precisely 10 groups of orders n and n + 1. %C A298428 Equivalently, lower member of consecutive terms of A249553. %H A298428 H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644. %H A298428 Gordon Royle, Numbers of Small Groups %H A298428 Index entries for sequences related to groups %F A298428 Sequence is { n | A000001(n) = 10, A000001(n+1) = 10 }. %e A298428 For n = 13914, A000001(13914) = A000001(13915) = 10. %e A298428 For n = 15974, A000001(15974) = A000001(15975) = 10. %e A298428 For n = 77234, A000001(77234) = A000001(77235) = 10. %p A298428 with(GroupTheory): for n from 1 to 10^5 do if [NumGroups(n), NumGroups(n+1)] = [10, 10] then print(n); fi; od; %Y A298428 Cf. A000001. Subsequence of A249553 (Numbers n having precisely 10 groups of order n). Numbers n having precisely k groups of orders n and n+1: A295230 (k=2), A295990 (k=4), A295991 (k=5), A295992 (k=6), A295993 (k=8), A298427 (k=9), this sequence (k=10), A295994 (k=11), A295995 (k=15). %K A298428 nonn,more,new %O A298428 1,1 %A A298428 _Muniru A Asiru_, Jan 19 2018 %I A298427 %S A298427 38227,113476,155827,269444,336931,411747 %N A298427 Numbers n such that there are precisely 9 groups of orders n and n + 1. %C A298427 Equivalently, lower member of consecutive terms of A249552. %H A298427 H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644. %H A298427 Gordon Royle, Numbers of Small Groups %H A298427 Index entries for sequences related to groups %F A298427 Sequence is { n | [A000001(n), A000001(n+1)] = [9, 9] }. %e A298427 For n = 38227, A000001(38227) = A000001(38228) = 9. %e A298427 For n = 113476, A000001(113476) = A000001(113477) = 9. %e A298427 For n = 155827, A000001(155827) = A000001(155828) = 9. %p A298427 with(GroupTheory): for n from 1 to 10^6 do if [NumGroups(n), NumGroups(n+1)] = [9, 9] then print(n); fi; od; %Y A298427 Cf. A000001. Subsequence of A249552 (Numbers n having precisely 9 groups of order n). Numbers n having precisely k groups of orders n and n+1: A295230 (k=2), A295990 (k=4), A295991 (k=5), A295992 (k=6), A295993 (k=8), this sequence (k=9), A298428 (k=10), A295994 (k=11), A295995 (k=15). %K A298427 nonn,more,new %O A298427 1,1 %A A298427 _Muniru A Asiru_, Jan 19 2018 %I A297970 %S A297970 112,240,368,496,624,752,880,1008,1136,1264,1392,1520,1648,1776,1904, %T A297970 2032,2160 %N A297970 Numbers that are not the sum of 3 squares and a nonnegative 7th power. %C A297970 The last term in this sequence is 2160. The reasons are as follows (let b, c, d, i, j, k, m, r, s, t, w, x, y and z be nonnegative integers). %C A297970 For the Diophantine equation x^2 + y^2 + z^2 + w^7 = m: %C A297970 (1) If m is not of the form 4^c * (8b + 7), then it follows from Legendre's three-square theorem that the equation has a solution with w = 0. %C A297970 (2) 8b + 7 - 1^7 = 8b + 6. Then m = 8b + 7, the equation has a solution with w = 1. %C A297970 (3) 4 * (8b + 7) - 1^7 = (8 * (4b + 3) + 3) = 8d + 3. Then m = 4 * (8b + 7), the equation has a solution with w = 1. %C A297970 (4) For b >= 17, 16 * (8b + 7) - 3^7 = 8 * (16 * (b - 17) + 12) + 5 = 8i + 5. Then m = 16 * (8b + 7) and b >= 17, the equation has a solution with w = 3. %C A297970 (5) 4^3 * (8b + 7) - 2^7 = 4^3 * (8b + 5). Then m = 4^3 * (8b + 7), the equation has a solution with w = 2. And 4^3 * (8b + 7) - 3^7 = 8 * (4^3 * (b - 4) + 38) + 5 = 8j + 5. Then m = 4^3 * (8b + 7) and b >= 4, the equation has a solution with w = 3. %C A297970 (6) 4^4 * (8b + 7) - 2^7 = 4^3 * (8 * (4b + 3) + 3) = 4^3 * (8k + 3). 4^4 * (8b + 7) - 3^7 = 8 * (256b - 217) + 3 = 8r + 3. Then m = 4^4 * (8b + 7), the equation has a solution with w = 2 and when b > 0, the equation has a solution with w = 3. %C A297970 (7) When c >= 5, 4^c * (8b + 7) - 2^7 = 4^3 * (8 * (b * 4^(c - 3) + 14 * 4^(c - 5) + 5) = 4^3 * (8s + 5). 4^c * (8b + 7) - 3^7 = 8 * (b * 4^(c - 3) + 14 * 4^(c - 3) - 273) + 3 = 8t + 3. Then n = 4^c * (8b + 7), the equation has solutions with w = 2 and 3. %C A297970 In short, except for the 17 numbers in the sequence, every nonnegative integer can be represented as the sum of 3 squares and a nonnegative 7th power. %H A297970 Wikipedia,Legendre's three-square theorem %F A297970 a(n) = 128n - 16 = 16 * A004771(n - 1), 1 <= n <= 17. %t A297970 t1={}; %t A297970 Do[Do[If[x^2+y^2+z^2+w^7==n, AppendTo[t1,n]&&Break[]], {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,y,(n-x^2-y^2)^(1/2)}, {w,0,(n-x^2-y^2-z^2)^(1/7)}], {n,0,3000}]; %t A297970 t2={}; %t A297970 Do[If[FreeQ[t1,k]==True, AppendTo[t2,k]], {k,0,3000}]; %t A297970 t2 %Y A297970 Finite subsequence of A004215 and A296185 %Y A297970 Cf. A004771, A022552, A022557, A022561, A022566, A111151. %K A297970 nonn,fini,full,new %O A297970 1,1 %A A297970 _XU Pingya_, Jan 10 2018 %I A297931 %S A297931 15,22,23,48,86,94,112,120,139,184,203,211,230,237,263,301,309,312, %T A297931 335,373,399,1014,1056,1455,1644,2029,2272,2658,3309,3469,4019,6502, %U A297931 11101 %N A297931 Numbers that are not the sum of a square and 3 nonnegative cubes. %C A297931 After 11101, there are no more terms up to 570000. %H A297931 Eric Weisstein's World of Mathematics, Waring's Problem %t A297931 t1={}; %t A297931 Do[Do[If[x^2+y^2+z^2+w^3==n, AppendTo[t1,n]&&Break[]], {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,y,(n-x^2-y^2)^(1/2)}, {w,0,(n-x^2-y^2-z^2)^(1/3)}], {n,0,5.7*10^5}]; %t A297931 t2={}; %t A297931 Do[If[FreeQ[t1,k]==True, AppendTo[t2,k]], {k,0,5.7*10^5}]; %t A297931 t2 %Y A297931 Cf. A004215, A022552, A022557, A022561, A022566, A111151. %K A297931 nonn,new %O A297931 1,1 %A A297931 _XU Pingya_, Jan 08 2018 %I A297930 %S A297930 1,2,3,2,2,2,2,1,2,4,5,3,2,3,2,1,3,5,6,3,3,3,2,0,2,5,6,5,4,5,2,2,4,5, %T A297930 6,4,6,6,4,2,4,6,4,4,4,7,3,2,4,3,5,4,7,8,5,3,3,3,5,5,5,6,4,3,6,7,8,7, %U A297930 5,7,4,2,7,9,10,4,5,7,3,3,9,10,8,5,4,7 %N A297930 Number of partitions of n into 2 squares and 2 nonnegative cubes. %C A297930 For n <= 5.45 * 10^5, except for a(23) = 0, all a(n) > 0. %C A297930 First occurrence of k beginning with 0: 23, 7, 1, 2, 9, 10, 18, 45, 53, 73, 74, 101, 125, 146, 165, 197, ..., . - _Robert G. Wilson v_, Jan 14 2018 %H A297930 Eric Weisstein's World of Mathematics, Waring's Problem %e A297930 2 = 0^2 + 0^2 + 1^3 + 1^3 = 0^2 + 1^2 + 0^3 + 1^3 = 1^2 + 1^2 + 0^3 + 0^3, a(2) = 3. %e A297930 10 = 0^2 + 1^2 + 1^3 + 2^3 = 0^2 + 3^2 + 0^3 + 1^3 = 1^2 + 1^2 + 0^3 + 2^3 = 1^2 + 3^2 + 0^3 + 0^3 = 2^2 + 2^2 + 1^3 + 1^3, a(10) = 5. %t A297930 a[n_]:=Sum[If[x^2+y^2+z^3+u^3==n,1,0], {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,0,(n-x^2-y^2)^(1/3)}, {u,z,(n-x^2-y^2-z^3)^(1/3)}] %t A297930 Table[a[n], {n,0,86}] %Y A297930 Cf. A000164, A002635, A274274. %K A297930 nonn,new %O A297930 0,2 %A A297930 _XU Pingya_, Jan 08 2018 %I A296602 %S A296602 4,19,23,25,29,31,33,35,39,41,43,45,49,51,53,55,57,59,61,63,65,67,69, %T A296602 71,73,75,77,79,81,83,85,87,89,91,93,95,97,99,101,103,105,107,109,111, %U A296602 113,115,117,119,121,123,125,127,129,131,133,135,137,139,141,143,145,147,149,151,153,155,157,159,161,163,165,167,169,171,173 %N A296602 Values of F for which there is a unique convex polyhedron with F faces that are all regular polygons. %C A296602 The main entry for this sequence is A180916. %C A296602 All terms except 4 are odd, because both the cube and the pentagonal pyramid have 6 faces, and for any even F > 6 both a prism and an antiprism can have F faces. Platonic solids, Archimedean solids, Johnson solids, and prisms account for the missing odd numbers. %F A296602 A180916(a(n)) = 1. %e A296602 The regular tetrahedron is the only convex polyhedron with 4 faces that are all regular polygons, and no such polyhedron with fewer than 4 faces exists, so a(1) = 4. %Y A296602 Cf. A180916, A242731, A296603, A296604. %K A296602 nonn,new %O A296602 1,1 %A A296602 _Jonathan Sondow_, Jan 28 2018 %I A298654 %S A298654 8,55,26,15,43,10,89,22,20,129,118,430,43,32,39,88,174,179,35,31,45, %T A298654 161,53,27,228,407,122,86,90,149,87,288,46,177,283,28,117,130,222,158, %U A298654 200,82,68,62,383,932,32,63,120,375,1107,67,298,110,119,352,122,277 %N A298654 Least number k such that the sum of the anti-divisors of k is equal to the sum of the anti-divisors of k+n. %e A298654 a(1) = 8 because the sum of the anti-divisors of 8 is 8 and of 9 is 8 again; %e A298654 a(2) = 55 because the sum of the anti-divisors of 55 is 74 and of 57 is 74 again. %p A298654 with(numtheory): P:=proc(q) local a,b,i,j,k,n; for i from 0 to q do for n from 1 to q do %p A298654 k:=0; j:=n; while j mod 2 <> 1 do k:=k+1; j:=j/2; od; %p A298654 a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2; %p A298654 k:=0; j:=n+i; while j mod 2 <> 1 do k:=k+1; j:=j/2; od; %p A298654 b:=sigma(2*(n+i)+1)+sigma(2*(n+i)-1)+sigma((n+i)/2^k)*2^(k+1)-6*(n+i)-2; %p A298654 if a=b then print(n); break; fi; od; od; end: P(10^5); %Y A298654 Cf. A007365, A066417. %K A298654 nonn,easy,new %O A298654 1,1 %A A298654 _Paolo P. Lava_, Jan 24 2018 %I A298483 %S A298483 13,25,37,61,73,109,113,117,121,153,157,169,173,181,245,257,273,277, %T A298483 285,289,297,313,317,325,333,353,361,369,373,385,389,401,405,409,421, %U A298483 425,457,509,513,525,529,541,601,609,621,637,653,673,677,693,705,709,729,733,761,765,769,777,797,801,805,829,833,841,853 %N A298483 Numbers n, the smallest of three consecutive numbers that share the property mu(n) <> chi(n). %C A298483 mu and chi share the same property in that they both evaluate to {-1, 0, 1}. %C A298483 This sequence admits 5 possible outcomes as follows: %C A298483 - a(n) are of the form 4k + 1, and are either divisible by an odd number of primes, or are nonsquarefree. %C A298483 - a(n) + 1 are squarefree even numbers. %C A298483 - a(n) + 2 are of the form 4k + 3, and are either divisible by an even number of primes, or are nonsquarefree. %C A298483 3 is the largest number of consecutive integers that satisfy the condition mu(n) <> chi(n). Since a(n) + 3 = 4k + 4 = 4(k+1), which is both nonsquarefree and even, then mu(4(k+1))= chi(4(k+1)), and the sequence terminates. %C A298483 If a(n) is prime then k - 2 is not divisible by 3. %C A298483 Conjecture: Every prime a(n) has a multiple a(m) + 1 m > n the result of multiplication by a squarefree even number, and has a multiple a(k) + 2 k > n the result of multiplication by a prime. Example; a(1) = 13, a(2) + 1 = 26, and a(3) + 2 = 39. %C A298483 If a(n) + 1 is a totient then k - 2 is not divisible by 3. %C A298483 Observation: Of the 72762 triples up to 10^6, only 19 of the middle terms, which are always even, are totients. %F A298483 0 < min({|mu(a(n))| + |chi(a(n))|, |mu(a(n) + 1)| + |chi(a(n) + 1)|, |mu(a(n) + 2)| + |chi(a(n) + 2)|}). %e A298483 13 is in the sequence because mu(13)=-1 and chi(13)=1, mu(14)=1 and chi(14)=0, and mu(15)=1 and chi(15)=-1. %o A298483 (PARI) isok(n) = (moebius(n) != kronecker( -4, n)) && (moebius(n+1) != kronecker( -4, n+1)) && (moebius(n+2) != kronecker( -4, n+2)); \\ _Michel Marcus_, Jan 28 2018 %Y A298483 Cf. A298482, A016813, A002144, A101455 (chi), A008683 (mu). %K A298483 nonn,new %O A298483 1,1 %A A298483 _Torlach Rush_, Jan 19 2018 %I A298856 %S A298856 3,10,21,55,78,105,136,171,253,351,406,465,595,666,741,820,903,1081, %T A298856 1275,1378,1711,1830,1953,2211,2485,2628,2775,2926,3081,3403,3741, %U A298856 3916,4465,4656,5050,5253,5671,5886,6105,6328,7021,7503,7750,8001,8515,9045,9316,9591 %N A298856 Triangular numbers n for which A240542(n) = A240542(n-1). %C A298856 Number n is in this sequence exactly when two parts of the symmetric representation of sigma(n) meet at the diagonal. %C A298856 Proof: If n = k*(2*k+1) is in this sequence then the length of row n in A240542 is 2*k and that of row n-1 is 2*k-1, i.e., the last leg of the Dyck path for n down to the diagonal is vertical and that for n-1 is horizontal to the same point on the diagonal. Therefore, one part of the symmetric representation of sigma(n) ends at the diagonal and so does its symmetric copy. Conversely, if two parts meet at the diagonal then the number of legs in the Dyck path to the diagonal for n, i.e., the length of row n in A240542, is one larger than that for n-1 and must be even, i.e., n has the form n = k*(2*k+1). %C A298856 A156592 is a subsequence since for every number of the form n = p*(2*p+1) where both p and 2*p+1 are primes A240542(n) = A240542(n-1). For a proof let T(n,k) = ceiling((n+1)/k - (k+1)/2) for 1 <= k <= floor((sqrt(8*n+1) - 1)/2) = 2*p, see A235791; then T(n,k) = T(n-1,k) + 1 for k = 1, 2, p, 2*p, and T(n,k) = T(n-1,k) for all other k. Therefore, the two alternating sums defining A240542(n) and A240542(n-1) are equal, i.e., their Dyck paths meet at the diagonal. %C A298856 Except for missing 10 the intersection of this sequence and A298855 equals A156592. Sequence A262259 is a subsequence of this sequence. %C A298856 The five known members of A191363 belong to this sequence, and since their symmetric representation consists of two parts of width one (the respective rows of triangle A237048 have the form 1 0 ... 0 1) they also belong to A262259. %e A298856 3, 10 and 21 are in the sequence as the illustration of Dyck paths in sequence A237048 shows. %e A298856 The sequence contains triangular numbers n*(2n+1) where neither n nor 2n+1 are prime. Numbers 1275=25*51 and 2926=38*77 are examples, however, 36 = 4*9 does not belong to the sequence. %e A298856 78 is the first number in the sequence whose two parts of its symmetric representation contain pieces of width two. %t A298856 (* Function path[] is defined in A237270 *) %t A298856 meetAtDiagonalQ[n_] := Module[{diags=Transpose[{Drop[Drop[path[n], 1], -1], path[n-1]}]}, Length[Union[diags[[n]]]]==1 && First[diags[[n-1]]]!=Last[diags[[n-1]]]] %t A298856 a298856[m_, n_] := Select[Map[#(2#+1)&, Range[m, n]], meetAtDiagonalQ] %t A298856 a298856[1, 70] (* data *) %Y A298856 Cf. A000217, A156592, A191363, A237048, A237270, A237593, A240542, A262259, A298855. %K A298856 nonn,new %O A298856 1,1 %A A298856 _Hartmut F. W. Hoft_, Jan 27 2018 %I A298855 %S A298855 21,33,39,51,55,57,65,69,85,87,93,95,111,115,119,123,129,133,141,145, %T A298855 155,159,161,177,183,185,201,203,205,213,215,217,219,235,237,249,253, %U A298855 259,265,267,287,291,295,301,303,305,309,319,321,327,329,335,339,341,355,365,371,377,381,393,395 %N A298855 Squarefree semiprimes p*q for which the symmetric representation of sigma(p*q) has four parts, in increasing order. %C A298855 All numbers in this sequence are odd since the symmetric representation of 2*p, p prime > 3, has two parts each of size 3*(p+1)/2, and that for 6 has one part of size 12. %C A298855 A number in this sequence has the form p*q, p and q prime, 3 <= p and 2*p < q, since in this case 2*p <= floor((sqrt(8*p*q + 1) - 1)/2) < q so that 1's in row p*q of A237048 occur only in positions 1, 2, p and 2*p. %C A298855 This sequence is a subsequence of A046388, hence of A006881, as well as of A174905, A241008 and A280107. %C A298855 The two central parts of the symmetric representation of sigma(p*q), each of size (p+q)/2, meet on the diagonal when q = 2*p + 1 since in this case 2*p = floor((sqrt(8*p*q + 1) - 1)/2). These triangular numbers p*(2p+1) form sequence A156592, except for its first element 10, and form a subsequence of the diagonal in the associated irregular triangle of this sequence given in the Example section. They also are a subsequence of A264104. A function to compute the coordinates on the diagonal where the two central parts meet is defined in sequence A240542. %C A298855 Except for missing 10 the intersection of this sequence and A298856 equals A156592. %e A298855 21=3*7 is the smallest number in the sequence since 2*3<7. %e A298855 1081=23*(2*23+1) is in the sequence; its central parts meet at 751 on the diagonal. %e A298855 The semiprimes p*q can be arranged as an irregular triangle with rows and columns labeled by the respective odd primes: %e A298855 q\p| 3 5 7 11 13 17 19 23 %e A298855 ---+--------------------------------------- %e A298855 7 | 21 %e A298855 11 | 33 55 %e A298855 13 | 39 65 %e A298855 17 | 51 85 119 %e A298855 19 | 57 95 133 %e A298855 23 | 69 115 161 253 %e A298855 29 | 87 145 203 319 377 %e A298855 31 | 93 155 217 341 403 %e A298855 37 | 111 185 259 407 481 629 %e A298855 41 | 123 205 287 451 533 697 779 %e A298855 43 | 129 215 301 473 559 731 817 %e A298855 47 | 141 235 329 517 611 799 893 1081 %t A298855 (* Function a237270[] is defined in A237270 *) %t A298855 a006881Q[n_] := Module[{f=FactorInteger[n]}, Length[f]==2 && AllTrue[Last[Transpose[f]], #==1&]] %t A298855 a298855[m_, n_] := Select[Range[m, n], a006881Q[#] && Length[a237270[#]]==4 &] %t A298855 a298855[1, 400] (* data *) %t A298855 (* column for prime p through number n *) %t A298855 stalk[n_, p_] := Select[a298855[1, n], First[First[FactorInteger[#]]]==p&] %Y A298855 Cf. A001358, A005384, A005385, A006881, A046388, A068443, A156592, A174905, A237048, A237270, A237593, A240542, A241008, A264104, A280107, A298856. %K A298855 nonn,tabf,new %O A298855 1,1 %A A298855 _Hartmut F. W. Hoft_, Jan 27 2018 %I A298753 %S A298753 1,0,1,0,1,1,0,2,3,1,0,5,8,7,1,0,14,23,26,15,1,0,42,70,89,80,31,1,0, %T A298753 132,222,302,335,242,63,1,0,429,726,1032,1294,1265,728,127,1,0,1430, %U A298753 2431,3564,4842,5654,4823,2186,255,1,0,4862,8294,12441,17886,23472,25270,18569,6560,511,1,0,16796,28730,43862,65767,93732 %N A298753 Triangle read by rows, T(n,m) = Sum_{k=1..m) k*k!*(-1)^(m+k)*Stirling2(m,k)* C(2*n+k-2*m-1,n-m)/(n+k-m), for n >= 0 and 0 <= m <= n. %F A298753 E.g.f.: 1/(1+C(x)*(exp(-x*y)-1)), where C(x)=A000108(x) is the g.f. of Catalan numbers. %F A298753 T(n,m) = Sum_{k=1..m) k*k!*(-1)^(m+k)*Stirling2(m,k)*C(2*n+k-2*m-1,n-m)/(n+k-m), mm, T(n,n)=1, E(n,m) is Euler triangle A008292. %e A298753 Triangle begins: %e A298753 1; %e A298753 0, 1; %e A298753 0, 1, 1; %e A298753 0, 2, 3, 1; %e A298753 0, 5, 8, 7, 1; %e A298753 0, 14, 23, 26, 15, 1; %e A298753 0, 42, 70, 89, 80, 31, 1; %e A298753 0, 132, 222, 302, 335, 242, 63, 1; %e A298753 0, 429, 726, 1032, 1294, 1265, 728, 127, 1; %p A298753 T := (n, m) -> `if`(n=0, 1, %p A298753 add(combinat[eulerian1](m, k-m)*binomial(2*n-k-1, n-k)*k/n, k=m..m+min(m,n-m))): %p A298753 for n from 0 to 8 do seq(T(n,k), k=0..n) od; # _Peter Luschny_, Jan 26 2018 %o A298753 (Maxima) %o A298753 T(n,m):=if n 1 such that (b^(2n) + 1)/2 has all prime divisors p == 1 (mod 2n). %C A298398 Conjecture: a(n) exists for every n. This is implied by the generalized Bunyakovsky conjecture (Schinzel's hypothesis H). %C A298398 The number (a(n)^(2n) + 1)/2 has all divisors d == 1 (mod 2n). %C A298398 Thus, here is the congruence a(n)^(2n) == 1 (mod 2n). %C A298398 If n is a power of 2, then a(n) = 3. %e A298398 a(5) = 9 and a(10) = 3 since (9^10 + 1)/2 = (3^20 + 1)/2 = 41 * 42521761. %p A298398 g:= proc(t) %p A298398 convert(select(type,map(s -> s[1], ifactors(t,easy)[2]),integer),set); %p A298398 end proc: %p A298398 F:= proc(n) local s,t,b,C,B,k,bb,Cb, easyf; uses numtheory; %p A298398 t:= 2^padic:-ordp(n,2); %p A298398 s:= n/t; %p A298398 C:= unapply({seq(numtheory:-cyclotomic(m,-b^(2*t)),m=numtheory:-divisors(s) minus {1}), (b^(2*t)+1)/2},b); %p A298398 B:= select(t -> C(t) mod (2*n) = {1}, [seq(b,b=1..2*n-1,2)]); %p A298398 for k from 0 do %p A298398 for bb in B do %p A298398 b:= k*2*n+bb; %p A298398 if b < 2 then next fi; %p A298398 Cb:= remove(isprime,C(b)); %p A298398 if Cb = {} then return b fi; %p A298398 easyf:= map(g, Cb) mod (2*n); %p A298398 if not (`union`(op(easyf)) subset {1}) then next fi; %p A298398 if andmap(c -> factorset(c) mod (2*n) = {1}, Cb) then return b fi; %p A298398 od %p A298398 od %p A298398 end proc: %p A298398 map(F, [$1..26]); # _Robert Israel_, Jan 18 2018 %t A298398 Array[Block[{b = 3}, While[Union@ Mod[FactorInteger[(b^(2 #) + 1)/2][[All, 1]], 2 #] != {1}, b += 2]; b] &, 20] (* _Michael De Vlieger_, Jan 20 2018 *) %t A298398 f[n_] := Block[{b = 3}, Label[init]; While[ PowerMod[b, 2n, 2n] != 1, b += 2]; d = First@# & /@ FactorInteger[(b^(2n) +1)/2]; If[ Union@ Mod[d, 2n] != {1}, b += 2; Goto[init]]; b]; Array[f, 30] (* _Robert G. Wilson v_, Jan 22 2018 *) %o A298398 (PARI) isok(b, n) = {pf = factor((b^(2*n) + 1)/2)[, 1]; for (j=1, #pf, if (lift(Mod(pf[j], 2*n)) != 1, return (0));); return(1);} %o A298398 a(n) = {my(b = 3); while (!isok(b, n), b += 2); b;} \\ _Michel Marcus_, Jan 19 2018 %Y A298398 Cf. A298299. %K A298398 nonn,hard,more,new %O A298398 1,1 %A A298398 _Thomas Ordowski_, Jan 18 2018 %E A298398 a(9)-a(30) from _Robert Israel_, Jan 18 2018 %E A298398 a(20) corrected by _Michel Marcus_, Jan 19 2018 %I A292364 %S A292364 4,8,9,12,24,121 %N A292364 Composites m such that each prime factor p > m of 2^m - 1 is a primitive prime factor of 2^m - 1. %C A292364 From A086251: "A prime factor of 2^n-1 is called primitive if it does not divide 2^r-1 for any r m of 2^m - 1. %o A292364 (PARI) lista(nn) = {forcomposite (m=1, nn, f = factor(2^m-1)[,1]~; ok = 1; for (k=1, #f, p = f[k]; if ((p > m) && (znorder(Mod(2, p)) != m), ok = 0; break);); if (ok, print1(m, ", ")););} \\ _Michel Marcus_, Nov 11 2017 %Y A292364 Cf. A002326, A060443, A086251. %K A292364 nonn,more,new %O A292364 1,1 %A A292364 _Thomas Ordowski_, Sep 15 2017 %I A298847 %S A298847 1,3,2,7,5,6,4,15,11,13,9,14,10,12,8,31,23,27,19,29,21,22,17,30,25,26, %T A298847 18,28,20,24,16,63,47,55,39,59,43,45,35,61,46,51,37,53,38,41,33,62,54, %U A298847 57,42,58,44,49,34,60,50,52,36,56,40,48,32,127,95,111,79 %N A298847 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, the number of ones in the binary expansion of n equals one plus the number of zeros in the binary expansion of a(n). %C A298847 In other words, for any n > 0, A000120(n) = 1 + A023416(a(n)). %C A298847 This sequence is a self-inverse permutation of the natural numbers, with fixed points A031448. %C A298847 We can build an analog of this sequence for any base b > 1: %C A298847 - let s_b be the sum of digits in base b, %C A298847 - in particular s_2 = A000120 and s_10 = A007953, %C A298847 - let l_b be the number of digits in base b, %C A298847 - in particular l_2 = A070939 and l_10 = A055642, %C A298847 - let f_b be the lexicographically earliest sequence of distinct positive terms such that, for any n > 0, s_b(n) = 1 + (b-1) * l_b(a(n)) - s_b(a(n)), %C A298847 - in particular, f_2 = a (this sequence), %C A298847 - f_b is a self-inverse permutation of the natural numbers, %C A298847 - l_b(n) = l_b(f_b(n)) for any n > 0, %C A298847 - f_b(b^k) = b^(k+1) - 1 for any k >= 0, %C A298847 - see also scatterplots of f_3 and f_10 in Links section. %H A298847 Rémy Sigrist, Table of n, a(n) for n = 1..8191 %H A298847 Rémy Sigrist, PARI program for A298847 %H A298847 Rémy Sigrist, Colored scatterplot of the first 2^16 - 1 terms (where the color is function of the Hamming weight of n) %H A298847 Rémy Sigrist, Scatterplot of the first 3^9 - 1 terms of f_3 %H A298847 Rémy Sigrist, Scatterplot of the first 10^4 - 1 terms of f_10 %H A298847 Index entries for sequences that are permutations of the natural numbers %F A298847 A070939(n) = A070939(a(n)) for any n > 0. %F A298847 a(2^k) = 2^(k+1) - 1 for any k >= 0. %F A298847 A000120(n) + A000120(a(n)) = 1 + A070939(n) for any n > 0. %e A298847 The first terms, alongside the binary representations of n and of a(n), are: %e A298847 n a(n) bin(n) bin(a(n)) %e A298847 -- ---- ------ --------- %e A298847 1 1 1 1 %e A298847 2 3 10 11 %e A298847 3 2 11 10 %e A298847 4 7 100 111 %e A298847 5 5 101 101 %e A298847 6 6 110 110 %e A298847 7 4 111 100 %e A298847 8 15 1000 1111 %e A298847 9 11 1001 1011 %e A298847 10 13 1010 1101 %e A298847 11 9 1011 1001 %e A298847 12 14 1100 1110 %e A298847 13 10 1101 1010 %e A298847 14 12 1110 1100 %e A298847 15 8 1111 1000 %o A298847 (PARI) See Links section. %Y A298847 Cf. A000120, A007953, A023416, A031448, A055642, A070939. %K A298847 nonn,base,new %O A298847 1,2 %A A298847 _Rémy Sigrist_, Jan 27 2018 %I A298859 %S A298859 1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, %T A298859 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, %U A298859 0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1 %N A298859 Number of partitions of n into distinct fourth powers. %H A298859 Eric Weisstein's World of Mathematics, Biquadratic Number %H A298859 Index entries for related partition-counting sequences %F A298859 G.f.: Product_{k>=1} (1 + x^(k^4)). %t A298859 nmax = 98; CoefficientList[Series[Product[1 + x^k^4, {k, 1, Floor[nmax^(1/4) + 1]}], {x, 0, nmax}], x] %Y A298859 Cf. A000009, A000583, A002377, A003999, A033461, A046039 (positions of zeros), A046042, A259793, A279329, A279485. %K A298859 nonn,new %O A298859 0 %A A298859 _Ilya Gutkovskiy_, Jan 27 2018 %I A298858 %S A298858 1,1,0,0,4,11,86,777,4670,36075,279482,2345201,21247326,197065752, %T A298858 1983741228,20769081251,228078253168,2604226354265,30880251148086, %U A298858 379415992755572,4818158748326064,63116999199457944,851467484377802094,11811530978240316682,168243449082524484856 %N A298858 Number of ordered ways of writing n-th triangular number as a sum of n nonzero triangular numbers. %H A298858 Eric Weisstein's World of Mathematics, Triangular Number %H A298858 Index to sequences related to polygonal numbers %F A298858 a(n) = [x^(n*(n+1)/2)] (Sum_{k>=1} x^(k*(k+1)/2))^n. %e A298858 a(4) = 4 because fourth triangular number is 10 and we have [3, 3, 3, 1], [3, 3, 1, 3], [3, 1, 3, 3] and [1, 3, 3, 3]. %t A298858 Table[SeriesCoefficient[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^n, {x, 0, n (n + 1)/2}], {n, 0, 24}] %Y A298858 Cf. A000217, A007294, A023361, A024940, A072964, A106337, A196010, A288126, A298329, A298330. %K A298858 nonn,new %O A298858 0,5 %A A298858 _Ilya Gutkovskiy_, Jan 27 2018 %I A298857 %S A298857 1,1,1,1,1,2,2,1,2,3,2,5,5,10,12,17,15,22,30,56,65,72,92,172,219,299, %T A298857 368,478,810,1055,1508,1778,2277,3815,5214,7103,8615,11614,18079, %U A298857 24428,33704,42877,56639,85597,116984,159179,199356,268965,400612,545674,740356,950897,1261597,1842307 %N A298857 Number of partitions of the n-th tetrahedral number into distinct tetrahedral numbers. %H A298857 Eric Weisstein's World of Mathematics, Tetrahedral Number %H A298857 Index to sequences related to pyramidal numbers %H A298857 Index entries for related partition-counting sequences %F A298857 a(n) = [x^A000292(n)] Product_{k>=1} (1 + x^A000292(k)). %F A298857 a(n) = A279278(A000292(n)). %e A298857 a(5) = 2 because fifth tetrahedral number is 35 and we have [35] and [20, 10, 4, 1]. %t A298857 Table[SeriesCoefficient[Product[1 + x^(k (k + 1) (k + 2)/6), {k, 1, n}], {x, 0, n (n + 1) (n + 2)/6}], {n, 0, 53}] %Y A298857 Cf. A000292, A030273, A279278, A288126, A298269. %K A298857 nonn,new %O A298857 0,6 %A A298857 _Ilya Gutkovskiy_, Jan 27 2018 %I A298848 %S A298848 1,0,1,1,1,1,2,1,1,1,1,1,3,2,5,4,3,4,13,11,15,20,23,34,52,49,97,118, %T A298848 164,192,296,330,525,745,825,1354,1820,1994,3356,4605,5543,8335,12319, %U A298848 13124,21133,28634,33209,51272,71154,85329,126806,174704,210157,310269,433490,511199 %N A298848 Number of partitions of n^3 into distinct cubes > 1. %H A298848 Index entries for sequences related to sums of cubes %H A298848 Index entries for related partition-counting sequences %F A298848 a(n) = [x^(n^3)] Product_{k>=2} (1 + x^(k^3)). %F A298848 a(n) = A280130(A000578(n)). %e A298848 a(6) = 2 because we have [216] and [125, 64, 27]. %Y A298848 Cf. A000578, A003108, A030272, A078128, A259792, A279329, A280130, A298641, A298671, A298672. %K A298848 nonn,new %O A298848 0,7 %A A298848 _Ilya Gutkovskiy_, Jan 27 2018 %I A298269 %S A298269 1,1,2,4,11,29,94,304,1005,3336,11398,38739,132340,451086,1541074, %T A298269 5242767,17779666,60048847,202124143,677000711,2256910444,7486274436, %U A298269 24713275977,81162110629,265192045408,862061443031,2788194736946,8972104829849,28726271274133,91515498561954,290116750935925 %N A298269 Number of partitions of the n-th tetrahedral number into tetrahedral numbers. %H A298269 Eric Weisstein's World of Mathematics, Tetrahedral Number %H A298269 Index to sequences related to pyramidal numbers %H A298269 Index entries for related partition-counting sequences %F A298269 a(n) = [x^A000292(n)] Product_{k>=1} 1/(1 - x^A000292(k)). %F A298269 a(n) = A068980(A000292(n)). %e A298269 a(3) = 4 because third tetrahedral number is 10 and we have [10], [4, 4, 1, 1], [4, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]. %t A298269 Table[SeriesCoefficient[Product[1/(1 - x^(k (k + 1) (k + 2)/6)), {k, 1, n}], {x, 0, n (n + 1) (n + 2)/6}], {n, 0, 30}] %Y A298269 Cf. A000292, A037444, A068980, A072964, A298857. %K A298269 nonn,new %O A298269 0,3 %A A298269 _Ilya Gutkovskiy_, Jan 27 2018 %I A298798 %S A298798 1,3,6,11,21,36,51,65,91,106,142 %N A298798 Coordination sequence for abf 3D net with respect to a trivalent node. %H A298798 Reticular Chemistry Structure Resource (RCSR), The abf 3D net %Y A298798 Cf. A298796, A298797. %K A298798 more,nonn,new %O A298798 0,2 %A A298798 _N. J. A. Sloane_, Jan 28 2018 %I A298797 %S A298797 1,4,8,12,20,48,44,60,98,120,116 %N A298797 Coordination sequence for abf 3D net with respect to a tetravalent node of the second type. %H A298797 Reticular Chemistry Structure Resource (RCSR), The abf 3D net %Y A298797 Cf. A298796, A298798. %K A298797 more,nonn,new %O A298797 0,2 %A A298797 _N. J. A. Sloane_, Jan 28 2018 %I A298796 %S A298796 1,4,4,12,24,36,40,84,82,100,148 %N A298796 Coordination sequence for abf 3D net with respect to a tetravalent node of the first type. %H A298796 Reticular Chemistry Structure Resource (RCSR), The abf 3D net %Y A298796 Cf. A298797, A298798. %K A298796 nonn,more,new %O A298796 0,2 %A A298796 _N. J. A. Sloane_, Jan 28 2018 %I A298795 %S A298795 1,3,6,8,11,16,19,18,23,30,29 %N A298795 Coordination sequence for the bil tiling (or net) with respect to a trivalent node of the fourth type. %H A298795 Reticular Chemistry Structure Resource (RCSR), The bil tiling (or net) %Y A298795 Cf. A298792, A298793, A298794. %K A298795 nonn,more,new %O A298795 0,2 %A A298795 _N. J. A. Sloane_, Jan 27 2018 %I A298794 %S A298794 1,3,6,7,11,16,19,21,21,26,33 %N A298794 Coordination sequence for the bil tiling (or net) with respect to a trivalent node of the third type. %H A298794 Reticular Chemistry Structure Resource (RCSR), The bil tiling (or net) %Y A298794 Cf. A298792, A298793, A298795. %K A298794 nonn,more,new %O A298794 0,2 %A A298794 _N. J. A. Sloane_, Jan 27 2018 %I A298793 %S A298793 1,3,6,8,12,15,17,21,24,27,29 %N A298793 Coordination sequence for the bil tiling (or net) with respect to a trivalent node of the second type. %H A298793 Reticular Chemistry Structure Resource (RCSR), The bil tiling (or net) %Y A298793 Cf. A298792, A298794, A298795. %K A298793 nonn,more,new %O A298793 0,2 %A A298793 _N. J. A. Sloane_, Jan 27 2018 %I A298902 %S A298902 0,0,0,0,1,0,0,1,1,0,0,2,1,2,0,0,3,2,2,3,0,0,5,3,4,3,5,0,0,8,7,10,10, %T A298902 7,8,0,0,13,12,24,32,24,12,13,0,0,21,25,56,120,120,56,25,21,0,0,34,47, %U A298902 142,471,963,471,142,47,34,0,0,55,96,346,2070,4689,4689,2070,346,96,55,0,0 %N A298902 T(n,k)=Number of nXk 0..1 arrays with every element equal to 3, 4, 5, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298902 Table starts %C A298902 .0..0..0...0....0......0.......0.........0..........0............0 %C A298902 .0..1..1...2....3......5.......8........13.........21...........34 %C A298902 .0..1..1...2....3......7......12........25.........47...........96 %C A298902 .0..2..2...4...10.....24......56.......142........346..........874 %C A298902 .0..3..3..10...32....120.....471......2070.......9055........39809 %C A298902 .0..5..7..24..120....963....4689.....34739.....206363......1388386 %C A298902 .0..8.12..56..471...4689...44186....522001....5379458.....62969638 %C A298902 .0.13.25.142.2070..34739..522001...9994726..165198014...3059725754 %C A298902 .0.21.47.346.9055.206363.5379458.165198014.4485120457.138682695325 %H A298902 R. H. Hardin, Table of n, a(n) for n = 1..220 %F A298902 Empirical for column k: %F A298902 k=1: a(n) = a(n-1) %F A298902 k=2: a(n) = a(n-1) +a(n-2) %F A298902 k=3: a(n) = 2*a(n-1) +a(n-2) -2*a(n-3) +a(n-4) -2*a(n-5) %F A298902 k=4: [order 16] %F A298902 k=5: [order 61] %e A298902 Some solutions for n=5 k=4 %e A298902 ..0..0..0..0. .0..0..1..1. .0..0..1..1. .0..0..1..1. .0..0..1..1 %e A298902 ..0..0..0..0. .0..0..1..1. .0..0..1..1. .0..0..1..1. .0..0..1..1 %e A298902 ..1..1..1..1. .0..0..1..1. .0..0..0..0. .1..1..1..1. .0..0..1..1 %e A298902 ..1..1..1..1. .0..0..1..1. .1..1..0..0. .1..1..0..0. .1..1..0..0 %e A298902 ..1..1..1..1. .0..0..1..1. .1..1..0..0. .1..1..0..0. .1..1..0..0 %Y A298902 Column 2 is A000045(n-1). %K A298902 nonn,tabl,new %O A298902 1,12 %A A298902 _R. H. Hardin_, Jan 28 2018 %I A298901 %S A298901 0,8,12,56,471,4689,44186,522001,5379458,62969638,689301117, %T A298901 7995632432,89669878606,1030894396261,11694235466700,133810754113662, %U A298901 1523947269885410,17410346219635865,198543925466100523 %N A298901 Number of nX7 0..1 arrays with every element equal to 3, 4, 5, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298901 Column 7 of A298902. %H A298901 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298901 Some solutions for n=5 %e A298901 ..0..0..0..0..0..1..1. .0..0..1..1..0..0..0. .0..0..0..0..0..1..1 %e A298901 ..0..0..1..0..0..1..1. .0..0..1..1..0..0..0. .0..0..0..0..0..1..1 %e A298901 ..0..1..1..1..1..1..1. .0..0..1..0..0..0..0. .0..0..0..0..0..0..1 %e A298901 ..0..0..1..1..1..0..0. .0..0..1..1..0..1..1. .1..1..1..1..0..1..1 %e A298901 ..0..0..1..1..1..0..0. .0..0..1..1..1..1..1. .1..1..1..1..1..1..1 %Y A298901 Cf. A298902. %K A298901 nonn,new %O A298901 1,2 %A A298901 _R. H. Hardin_, Jan 28 2018 %I A298900 %S A298900 0,5,7,24,120,963,4689,34739,206363,1388386,8770132,58277644, %T A298900 374054863,2461101248,15971848701,104543429551,680910307295, %U A298900 4450730822243,29026246222317,189592201754404,1237265914602821,8078981295508349 %N A298900 Number of nX6 0..1 arrays with every element equal to 3, 4, 5, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298900 Column 6 of A298902. %H A298900 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298900 Some solutions for n=5 %e A298900 ..0..0..0..0..0..0. .0..0..1..1..0..0. .0..0..0..0..0..0. .0..0..0..1..1..1 %e A298900 ..0..0..1..1..0..0. .0..0..1..1..0..0. .0..0..1..0..0..0. .0..0..0..1..1..1 %e A298900 ..1..1..0..1..1..1. .0..0..0..1..1..1. .0..1..1..1..1..1. .1..1..1..1..1..1 %e A298900 ..1..1..0..0..1..1. .1..1..0..0..1..1. .0..0..1..1..1..1. .1..1..1..0..0..0 %e A298900 ..1..1..0..0..1..1. .1..1..0..0..1..1. .0..0..1..1..1..1. .1..1..1..0..0..0 %Y A298900 Cf. A298902. %K A298900 nonn,new %O A298900 1,2 %A A298900 _R. H. Hardin_, Jan 28 2018 %I A298899 %S A298899 0,3,3,10,32,120,471,2070,9055,39809,178728,805614,3628936,16427863, %T A298899 74247523,336094409,1520856812,6885288837,31164948670,141091227797, %U A298899 638704784849,2891500909471,13090012145416,59260093885255 %N A298899 Number of nX5 0..1 arrays with every element equal to 3, 4, 5, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298899 Column 5 of A298902. %H A298899 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298899 Empirical: a(n) = 3*a(n-1) +14*a(n-2) -10*a(n-3) -74*a(n-4) -174*a(n-5) +92*a(n-6) +550*a(n-7) +284*a(n-8) +2822*a(n-9) +387*a(n-10) -4568*a(n-11) +667*a(n-12) -34031*a(n-13) +14309*a(n-14) -20283*a(n-15) -15631*a(n-16) +126141*a(n-17) -18534*a(n-18) +138037*a(n-19) -92804*a(n-20) -62863*a(n-21) -145683*a(n-22) +145669*a(n-23) -252735*a(n-24) +856599*a(n-25) +603305*a(n-26) -951928*a(n-27) +40836*a(n-28) -1648325*a(n-29) +67746*a(n-30) -2832702*a(n-31) +2920682*a(n-32) +2347923*a(n-33) +2455740*a(n-34) -529807*a(n-35) -3152693*a(n-36) -1484407*a(n-37) -780449*a(n-38) -47439*a(n-39) -945802*a(n-40) +5691923*a(n-41) +2876338*a(n-42) -4570720*a(n-43) -3088139*a(n-44) +889471*a(n-45) +1193058*a(n-46) +289889*a(n-47) +642191*a(n-48) -399302*a(n-49) -564706*a(n-50) +60480*a(n-51) +33698*a(n-52) +64879*a(n-53) +81675*a(n-54) -1730*a(n-55) -6011*a(n-56) -1096*a(n-57) -7487*a(n-58) -2317*a(n-59) -530*a(n-60) -144*a(n-61) %e A298899 Some solutions for n=5 %e A298899 ..0..0..0..1..1. .0..0..0..1..1. .0..0..1..1..1. .0..0..1..1..1 %e A298899 ..0..0..0..1..1. .0..0..0..1..1. .0..0..1..1..1. .0..0..1..1..1 %e A298899 ..0..0..0..0..1. .0..0..0..0..1. .0..0..1..1..1. .0..0..1..1..1 %e A298899 ..0..0..0..1..1. .1..1..0..1..1. .1..1..0..0..0. .0..0..1..1..1 %e A298899 ..0..0..0..1..1. .1..1..1..1..1. .1..1..0..0..0. .0..0..1..1..1 %Y A298899 Cf. A298902. %K A298899 nonn,new %O A298899 1,2 %A A298899 _R. H. Hardin_, Jan 28 2018 %I A298898 %S A298898 0,2,2,4,10,24,56,142,346,874,2210,5640,14440,37224,96128,249108, %T A298898 646710,1681772,4378614,11411374,29760404,77658384,202729956, %U A298898 529406180,1382821596,3612643922,9439431124,24666860228,64464076954,168480273482 %N A298898 Number of nX4 0..1 arrays with every element equal to 3, 4, 5, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298898 Column 4 of A298902. %H A298898 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298898 Empirical: a(n) = a(n-1) +5*a(n-2) +4*a(n-3) -6*a(n-4) -18*a(n-5) -16*a(n-6) -12*a(n-7) -3*a(n-8) +9*a(n-9) +22*a(n-10) +19*a(n-11) +14*a(n-12) +3*a(n-13) -a(n-14) -2*a(n-15) -2*a(n-16) %e A298898 Some solutions for n=5 %e A298898 ..0..0..1..1. .0..0..0..0. .0..0..0..0. .0..0..1..1. .0..0..1..1 %e A298898 ..0..0..1..1. .0..0..0..0. .0..0..0..0. .0..0..1..1. .0..0..1..1 %e A298898 ..1..1..1..1. .0..0..0..0. .0..0..0..0. .1..1..0..0. .0..1..1..1 %e A298898 ..1..1..0..0. .0..0..0..0. .1..1..1..1. .1..1..0..0. .0..0..1..1 %e A298898 ..1..1..0..0. .0..0..0..0. .1..1..1..1. .1..1..0..0. .0..0..1..1 %Y A298898 Cf. A298902. %K A298898 nonn,new %O A298898 1,2 %A A298898 _R. H. Hardin_, Jan 28 2018 %I A298897 %S A298897 0,1,1,2,3,7,12,25,47,96,187,377,746,1497,2981,5970,11919,23851,47668, %T A298897 95357,190659,381352,762615,1525285,3050426,6100941,12201649,24403442, %U A298897 48806507,97613247,195225884,390452145,780903303,1561807216,3123612835 %N A298897 Number of nX3 0..1 arrays with every element equal to 3, 4, 5, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298897 Column 3 of A298902. %H A298897 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298897 Empirical: a(n) = 2*a(n-1) +a(n-2) -2*a(n-3) +a(n-4) -2*a(n-5) %e A298897 All solutions for n=5 %e A298897 ..0..0..0. .0..0..0. .0..0..0 %e A298897 ..0..0..0. .0..0..0. .0..0..0 %e A298897 ..0..0..0. .1..1..1. .0..0..0 %e A298897 ..0..0..0. .1..1..1. .1..1..1 %e A298897 ..0..0..0. .1..1..1. .1..1..1 %Y A298897 Cf. A298902. %K A298897 nonn,new %O A298897 1,4 %A A298897 _R. H. Hardin_, Jan 28 2018 %I A298896 %S A298896 0,1,1,4,32,963,44186,9994726,4485120457,6962969227448 %N A298896 Number of nXn 0..1 arrays with every element equal to 3, 4, 5, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298896 Diagonal of A298902. %e A298896 Some solutions for n=5 %e A298896 ..0..0..1..1..1. .0..0..1..1..1. .0..0..0..1..1. .0..0..0..1..1 %e A298896 ..0..0..1..1..1. .0..0..1..1..1. .0..0..0..1..1. .0..0..0..1..1 %e A298896 ..0..1..1..1..1. .1..1..1..1..1. .1..1..0..1..1. .1..1..1..0..0 %e A298896 ..0..0..1..1..1. .1..1..1..0..0. .1..1..0..0..0. .1..1..1..0..0 %e A298896 ..0..0..1..1..1. .1..1..1..0..0. .1..1..0..0..0. .1..1..1..0..0 %Y A298896 Cf. A298902. %K A298896 nonn,new %O A298896 1,4 %A A298896 _R. H. Hardin_, Jan 28 2018 %I A298895 %S A298895 0,0,0,0,1,0,0,3,3,0,0,2,1,2,0,0,11,4,4,11,0,0,13,4,11,4,13,0,0,34,11, %T A298895 31,31,11,34,0,0,65,26,80,219,80,26,65,0,0,123,66,229,579,579,229,66, %U A298895 123,0,0,266,171,681,1858,2963,1858,681,171,266,0,0,499,462,1969,8891,12224 %N A298895 T(n,k)=Number of nXk 0..1 arrays with every element equal to 2, 3, 5, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298895 Table starts %C A298895 .0...0...0....0.....0......0.......0........0.........0..........0............0 %C A298895 .0...1...3....2....11.....13......34.......65.......123........266..........499 %C A298895 .0...3...1....4.....4.....11......26.......66.......171........462.........1248 %C A298895 .0...2...4...11....31.....80.....229......681......1969.......5973........18031 %C A298895 .0..11...4...31...219....579....1858.....8891.....34212.....128103.......538967 %C A298895 .0..13..11...80...579...2963...12224....72620....426475....2284203.....12768382 %C A298895 .0..34..26..229..1858..12224...74725...547497...4012035...28805843....209118279 %C A298895 .0..65..66..681..8891..72620..547497..5719782..54423404..502721390...4804572705 %C A298895 .0.123.171.1969.34212.426475.4012035.54423404.724538480.8857192378.112203393143 %H A298895 R. H. Hardin, Table of n, a(n) for n = 1..219 %F A298895 Empirical for column k: %F A298895 k=1: a(n) = a(n-1) %F A298895 k=2: a(n) = a(n-1) +3*a(n-2) -4*a(n-4) %F A298895 k=3: [order 17] for n>18 %F A298895 k=4: [order 58] for n>59 %e A298895 Some solutions for n=7 k=4 %e A298895 ..0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..0..0. .0..0..0..0 %e A298895 ..0..0..0..0. .0..1..0..1. .0..0..0..0. .0..0..0..0. .0..0..0..0 %e A298895 ..1..1..1..1. .0..1..1..0. .0..0..0..0. .0..0..0..0. .0..0..0..0 %e A298895 ..1..1..1..1. .0..0..0..0. .0..0..0..0. .1..1..1..1. .1..1..1..1 %e A298895 ..0..0..0..0. .1..1..1..1. .0..0..0..0. .1..0..0..1. .1..0..0..1 %e A298895 ..0..0..0..0. .1..1..1..1. .1..1..1..1. .0..1..0..1. .0..1..1..0 %e A298895 ..0..0..0..0. .1..1..1..1. .1..1..1..1. .0..0..1..1. .0..0..0..0 %Y A298895 Column 2 is A297870. %K A298895 nonn,tabl,new %O A298895 1,8 %A A298895 _R. H. Hardin_, Jan 28 2018 %I A298894 %S A298894 0,34,26,229,1858,12224,74725,547497,4012035,28805843,209118279, %T A298894 1537327380,11240705948,82185225848,602873376262,4421261929905, %U A298894 32410993796867,237712768537660,1743828733781802,12791372365817836 %N A298894 Number of nX7 0..1 arrays with every element equal to 2, 3, 5, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298894 Column 7 of A298895. %H A298894 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298894 Some solutions for n=7 %e A298894 ..0..0..1..1..0..0..0. .0..0..0..1..1..1..1. .0..0..1..0..0..1..1 %e A298894 ..0..1..0..0..1..1..0. .0..0..0..1..1..1..1. .0..1..1..0..0..1..1 %e A298894 ..0..1..0..1..1..1..0. .1..1..1..1..1..1..1. .1..0..0..1..0..0..0 %e A298894 ..0..1..0..1..0..0..0. .1..1..1..1..0..0..0. .1..1..1..1..0..0..0 %e A298894 ..0..1..0..1..0..1..0. .1..1..1..1..0..0..0. .1..0..1..1..1..1..1 %e A298894 ..1..0..0..1..0..1..1. .1..1..1..1..0..0..0. .1..0..0..1..1..1..1 %e A298894 ..1..1..1..1..0..0..1. .1..1..1..1..0..0..0. .1..1..0..1..1..1..1 %Y A298894 Cf. A298895. %K A298894 nonn,new %O A298894 1,2 %A A298894 _R. H. Hardin_, Jan 28 2018 %I A298893 %S A298893 0,13,11,80,579,2963,12224,72620,426475,2284203,12768382,72904621, %T A298893 409673880,2298677918,13008113489,73461302729,414141245069, %U A298893 2338885431088,13210876247129,74582366744334,421146376787631,2378532739728979 %N A298893 Number of nX6 0..1 arrays with every element equal to 2, 3, 5, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298893 Column 6 of A298895. %H A298893 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298893 Some solutions for n=7 %e A298893 ..0..0..0..0..1..1. .0..0..1..0..0..1. .0..0..0..1..1..0. .0..0..0..1..1..1 %e A298893 ..0..1..1..1..0..1. .0..1..1..0..1..1. .0..0..0..1..0..0. .0..0..0..1..1..1 %e A298893 ..1..0..1..0..0..1. .0..1..0..0..0..0. .1..1..1..1..1..1. .1..1..1..1..1..1 %e A298893 ..0..1..0..1..1..1. .1..0..0..0..0..0. .1..1..1..1..1..1. .1..1..1..0..0..0 %e A298893 ..0..0..0..0..0..0. .0..1..0..0..0..0. .1..1..1..1..1..1. .1..1..1..0..0..0 %e A298893 ..0..1..0..0..0..0. .0..1..1..0..1..1. .1..1..1..1..1..1. .0..0..1..1..1..1 %e A298893 ..1..1..0..0..0..0. .0..0..1..0..0..1. .1..1..1..1..1..1. .0..0..1..1..1..1 %Y A298893 Cf. A298895. %K A298893 nonn,new %O A298893 1,2 %A A298893 _R. H. Hardin_, Jan 28 2018 %I A298892 %S A298892 0,11,4,31,219,579,1858,8891,34212,128103,538967,2219296,8764075, %T A298892 36216096,148454907,602014960,2469409541,10115533795,41307139499, %U A298892 169152390705,692711105743,2833870255179,11601723412009,47502652641814 %N A298892 Number of nX5 0..1 arrays with every element equal to 2, 3, 5, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298892 Column 5 of A298895. %H A298892 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298892 Some solutions for n=7 %e A298892 ..0..0..0..1..1. .0..0..1..0..0. .0..0..1..1..0. .0..0..1..1..1 %e A298892 ..0..0..0..1..1. .0..1..1..1..0. .0..1..1..0..0. .0..0..1..1..1 %e A298892 ..1..1..0..0..0. .0..1..0..1..0. .1..1..1..1..1. .0..0..1..1..1 %e A298892 ..0..1..0..0..0. .0..0..1..0..0. .0..0..1..1..1. .1..1..1..0..0 %e A298892 ..0..0..0..0..0. .1..1..1..1..1. .0..0..1..1..1. .1..1..0..1..0 %e A298892 ..0..1..0..0..0. .1..1..1..0..1. .0..0..1..1..1. .1..1..0..1..1 %e A298892 ..1..1..0..0..0. .1..1..1..0..0. .0..0..1..1..1. .1..1..0..0..1 %Y A298892 Cf. A298895. %K A298892 nonn,new %O A298892 1,2 %A A298892 _R. H. Hardin_, Jan 28 2018 %I A298891 %S A298891 0,2,4,11,31,80,229,681,1969,5973,18031,54874,167752,513625,1575095, %T A298891 4835994,14859480,45674676,140452387,431987865,1328865560,4088283165, %U A298891 12578581331,38703117525,119090222855,366452309799,1127630050984 %N A298891 Number of nX4 0..1 arrays with every element equal to 2, 3, 5, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298891 Column 4 of A298895. %H A298891 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298891 Empirical: a(n) = 6*a(n-1) -9*a(n-2) +5*a(n-3) -32*a(n-4) +38*a(n-5) +25*a(n-6) +98*a(n-7) -119*a(n-8) +40*a(n-9) -550*a(n-10) +580*a(n-11) -916*a(n-12) +830*a(n-13) +263*a(n-14) +83*a(n-15) +2328*a(n-16) -229*a(n-17) +503*a(n-18) -4047*a(n-19) -417*a(n-20) -3044*a(n-21) +2435*a(n-22) -11065*a(n-23) +3356*a(n-24) -5303*a(n-25) +1559*a(n-26) +14545*a(n-27) +32995*a(n-28) +22512*a(n-29) -15204*a(n-30) -12820*a(n-31) -31577*a(n-32) -21644*a(n-33) +3315*a(n-34) +18307*a(n-35) -24537*a(n-36) -14373*a(n-37) +11731*a(n-38) +26599*a(n-39) +15792*a(n-40) -11544*a(n-41) -3386*a(n-42) -4977*a(n-43) +5179*a(n-44) -2851*a(n-45) +4056*a(n-46) -977*a(n-47) +3855*a(n-48) -121*a(n-49) +305*a(n-50) -1567*a(n-51) -458*a(n-52) -273*a(n-53) +321*a(n-54) -8*a(n-55) +78*a(n-56) -20*a(n-57) -6*a(n-58) for n>59 %e A298891 Some solutions for n=7 %e A298891 ..0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..1..1 %e A298891 ..0..0..0..0. .0..0..0..0. .0..0..0..0. .0..1..1..0. .0..1..0..1 %e A298891 ..0..0..0..0. .0..0..0..0. .0..0..0..0. .1..0..0..1. .1..0..0..1 %e A298891 ..1..1..1..1. .1..1..1..1. .0..0..0..0. .1..1..1..1. .1..1..1..1 %e A298891 ..1..0..0..1. .1..1..1..1. .0..0..0..0. .0..0..0..0. .0..0..0..0 %e A298891 ..0..1..1..0. .0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..0 %e A298891 ..0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..0 %Y A298891 Cf. A298895. %K A298891 nonn,new %O A298891 1,2 %A A298891 _R. H. Hardin_, Jan 28 2018 %I A298890 %S A298890 0,3,1,4,4,11,26,66,171,462,1248,3419,9450,26334,73697,206960,582316, %T A298890 1640549,4625476,13047636,36816651,103906694,293290860,827923703, %U A298890 2337253142,6598367806,18628473233,52592572696,148482655256,419208157101 %N A298890 Number of nX3 0..1 arrays with every element equal to 2, 3, 5, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298890 Column 3 of A298895. %H A298890 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298890 Empirical: a(n) = 4*a(n-1) -a(n-2) -5*a(n-3) -10*a(n-4) +11*a(n-5) +15*a(n-6) +2*a(n-7) -5*a(n-8) -33*a(n-9) -20*a(n-10) +47*a(n-11) +36*a(n-12) +4*a(n-14) -12*a(n-15) -2*a(n-16) +2*a(n-17) for n>18 %e A298890 Some solutions for n=7 %e A298890 ..0..0..0. .0..0..1. .0..0..0. .0..0..0. .0..1..1. .0..0..0. .0..1..1 %e A298890 ..0..0..0. .0..1..1. .0..0..0. .0..0..0. .0..0..1. .1..0..1. .0..0..1 %e A298890 ..1..1..1. .1..0..0. .0..0..0. .1..1..1. .1..0..1. .1..1..1. .1..0..1 %e A298890 ..0..1..0. .1..0..1. .0..0..0. .1..1..1. .1..1..0. .0..0..1. .1..1..0 %e A298890 ..0..0..0. .1..1..1. .1..1..1. .0..0..0. .1..0..1. .0..1..0. .0..0..0 %e A298890 ..0..1..0. .1..0..1. .1..1..1. .0..0..0. .0..0..1. .1..1..0. .0..1..0 %e A298890 ..1..1..1. .0..0..0. .1..1..1. .0..0..0. .0..1..1. .1..0..0. .1..1..1 %Y A298890 Cf. A298895. %K A298890 nonn,new %O A298890 1,2 %A A298890 _R. H. Hardin_, Jan 28 2018 %I A298889 %S A298889 0,1,1,11,219,2963,74725,5719782,724538480,145951109194 %N A298889 Number of nXn 0..1 arrays with every element equal to 2, 3, 5, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298889 Diagonal of A298895. %e A298889 Some solutions for n=7 %e A298889 ..0..0..0..0..1..1..1. .0..0..0..1..1..1..0. .0..0..1..0..0..1..1 %e A298889 ..0..1..1..0..1..1..1. .0..1..1..0..1..0..0. .1..0..1..1..0..0..1 %e A298889 ..1..0..1..1..1..0..0. .1..0..0..1..1..1..0. .1..1..0..0..0..0..0 %e A298889 ..1..0..0..0..1..1..0. .1..0..1..1..1..1..1. .0..0..0..0..1..1..0 %e A298889 ..1..0..1..0..0..0..1. .0..1..1..1..0..0..1. .0..0..0..1..0..1..0 %e A298889 ..1..0..1..1..1..1..0. .1..0..1..0..1..1..0. .0..0..0..1..0..0..1 %e A298889 ..1..1..0..0..0..0..0. .1..1..0..1..0..0..0. .0..0..0..1..1..1..1 %Y A298889 Cf. A298895. %K A298889 nonn,new %O A298889 1,4 %A A298889 _R. H. Hardin_, Jan 28 2018 %I A298888 %S A298888 0,1,1,1,3,1,2,7,7,2,3,13,15,13,3,5,23,19,19,23,5,8,49,23,40,23,49,8, %T A298888 13,95,34,73,73,34,95,13,21,177,63,141,123,141,63,177,21,34,359,96, %U A298888 240,243,243,240,96,359,34,55,705,147,428,444,516,444,428,147,705,55,89,1351,233 %N A298888 T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 4, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298888 Table starts %C A298888 ..0...1...1...2....3....5....8....13....21....34.....55.....89.....144.....233 %C A298888 ..1...3...7..13...23...49...95...177...359...705...1351...2689....5303...10321 %C A298888 ..1...7..15..19...23...34...63....96...147...233....368....588.....933....1500 %C A298888 ..2..13..19..40...73..141..240...428...779..1531...2989...5729...10760...20205 %C A298888 ..3..23..23..73..123..243..444...897..1801..3462...6669..13291...25762...49483 %C A298888 ..5..49..34.141..243..516..814..1646..3312..6565..13040..25941...51679..103895 %C A298888 ..8..95..63.240..444..814.1818..3663..7496.16544..33596..70466..148451..311053 %C A298888 .13.177..96.428..897.1646.3663..7768.17225.39617..83996.184024..411042..903972 %C A298888 .21.359.147.779.1801.3312.7496.17225.41048.95996.221358.505218.1183870.2737277 %H A298888 R. H. Hardin, Table of n, a(n) for n = 1..312 %F A298888 Empirical for column k: %F A298888 k=1: a(n) = a(n-1) +a(n-2) %F A298888 k=2: a(n) = 3*a(n-1) -2*a(n-2) +4*a(n-3) -10*a(n-4) +4*a(n-5) for n>6 %F A298888 k=3: [order 18] for n>19 %F A298888 k=4: [order 72] for n>73 %e A298888 Some solutions for n=7 k=5 %e A298888 ..0..1..0..0..0. .0..1..0..1..1. .0..0..1..0..1. .0..1..1..0..1 %e A298888 ..0..0..1..1..1. .1..0..0..1..0. .1..0..1..1..0. .1..0..0..0..1 %e A298888 ..0..0..0..0..0. .0..1..1..1..0. .1..0..0..0..1. .1..0..0..0..1 %e A298888 ..1..0..0..0..1. .0..1..1..1..0. .1..0..0..0..1. .0..0..0..0..0 %e A298888 ..1..0..0..0..1. .1..1..1..1..1. .0..0..0..0..0. .1..1..1..0..0 %e A298888 ..0..1..1..0..1. .1..1..0..0..0. .1..1..1..0..0. .0..0..0..0..1 %e A298888 ..1..0..1..0..0. .1..0..1..1..1. .0..0..0..1..0. .1..1..1..0..1 %Y A298888 Column 1 is A000045(n-1). %Y A298888 Column 2 is A297852. %Y A298888 Column 3 is A298050. %Y A298888 Column 4 is A298051. %K A298888 nonn,tabl,new %O A298888 1,5 %A A298888 _R. H. Hardin_, Jan 28 2018 %I A298887 %S A298887 8,95,63,240,444,814,1818,3663,7496,16544,33596,70466,148451,311053, %T A298887 653252,1372584,2873709,6036087,12668750,26606071,55876191,117364939, %U A298887 246515244,517702247,1087644299,2284968402,4799996646,10083584000 %N A298887 Number of nX7 0..1 arrays with every element equal to 1, 2, 4, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298887 Column 7 of A298888. %H A298887 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298887 Some solutions for n=7 %e A298887 ..0..1..0..1..0..0..1. .0..0..0..1..1..1..0. .0..0..1..0..0..1..1 %e A298887 ..1..0..0..1..1..0..1. .1..0..0..0..0..0..1. .1..0..1..1..1..0..0 %e A298887 ..0..1..1..1..0..1..0. .0..1..0..0..0..1..0. .1..0..0..0..1..1..1 %e A298887 ..0..1..1..1..1..1..0. .0..1..0..0..0..1..0. .1..0..0..0..0..1..0 %e A298887 ..1..1..1..1..0..1..1. .1..0..0..0..0..0..1. .0..0..0..0..1..0..0 %e A298887 ..0..0..0..1..1..0..0. .1..0..1..0..1..0..1. .1..1..1..0..0..1..1 %e A298887 ..1..1..1..0..0..1..0. .0..0..1..0..1..0..0. .0..0..0..1..1..0..1 %Y A298887 Cf. A298888. %K A298887 nonn,new %O A298887 1,1 %A A298887 _R. H. Hardin_, Jan 28 2018 %I A298886 %S A298886 5,49,34,141,243,516,814,1646,3312,6565,13040,25941,51679,103895, %T A298886 205265,411011,821599,1643914,3279755,6573928,13136700,26293574, %U A298886 52604464,105288984,210671989,421821199,844171966,1689881070,3382803573 %N A298886 Number of nX6 0..1 arrays with every element equal to 1, 2, 4, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298886 Column 6 of A298888. %H A298886 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298886 Some solutions for n=7 %e A298886 ..0..1..0..0..1..1. .0..1..1..1..0..1. .0..1..1..0..0..0. .0..0..0..1..1..0 %e A298886 ..1..0..0..0..0..0. .1..0..0..0..0..1. .0..0..0..1..1..1. .1..0..0..0..0..1 %e A298886 ..0..0..0..1..0..1. .0..1..0..0..0..1. .1..1..0..0..0..0. .0..1..0..0..0..1 %e A298886 ..1..1..1..1..0..1. .0..1..0..0..0..0. .0..0..0..0..0..1. .0..1..0..0..0..0 %e A298886 ..0..1..1..1..0..1. .0..1..0..1..1..1. .1..1..0..0..0..1. .0..1..0..1..1..1 %e A298886 ..0..1..1..1..1..0. .1..1..1..1..1..0. .0..0..0..1..1..0. .1..1..1..1..1..0 %e A298886 ..1..0..0..1..1..1. .0..0..1..1..0..1. .1..1..0..1..0..1. .0..0..1..1..0..1 %Y A298886 Cf. A298888. %K A298886 nonn,new %O A298886 1,1 %A A298886 _R. H. Hardin_, Jan 28 2018 %I A298885 %S A298885 3,23,23,73,123,243,444,897,1801,3462,6669,13291,25762,49483,97860, %T A298885 193033,376031,735990,1440269,2815283,5522803,10827092,21196028, %U A298885 41505035,81298779,159266998,311969014,611140770,1197086348,2345047342,4593936013 %N A298885 Number of nX5 0..1 arrays with every element equal to 1, 2, 4, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298885 Column 5 of A298888. %H A298885 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298885 Some solutions for n=7 %e A298885 ..0..1..0..0..1. .0..0..0..1..0. .0..0..0..1..0. .0..1..1..0..1 %e A298885 ..0..1..1..1..0. .1..1..1..0..0. .1..1..1..1..0. .1..0..0..0..1 %e A298885 ..0..1..1..1..0. .0..0..0..0..0. .0..1..1..1..0. .1..0..0..0..1 %e A298885 ..1..1..1..1..1. .1..0..0..0..1. .0..1..1..1..1. .0..0..0..0..0 %e A298885 ..0..0..0..1..1. .1..0..0..0..1. .1..0..0..1..1. .1..1..1..0..0 %e A298885 ..1..1..1..1..0. .1..0..1..1..0. .0..0..1..1..0. .0..0..0..0..1 %e A298885 ..0..0..0..1..0. .0..0..1..0..1. .1..1..0..1..0. .1..1..1..0..1 %Y A298885 Cf. A298888. %K A298885 nonn,new %O A298885 1,1 %A A298885 _R. H. Hardin_, Jan 28 2018 %I A298884 %S A298884 0,3,15,40,123,516,1818,7768,41048,242083,1483959,10171555 %N A298884 Number of nXn 0..1 arrays with every element equal to 1, 2, 4, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298884 Diagonal of A298888. %e A298884 Some solutions for n=7 %e A298884 ..0..1..0..1..0..0..0. .0..1..1..0..0..1..1. .0..0..1..0..0..1..1 %e A298884 ..1..0..0..1..1..1..1. .1..0..0..0..0..0..0. .1..0..1..1..1..0..0 %e A298884 ..0..0..0..1..1..1..0. .1..0..0..0..1..0..1. .0..1..0..1..0..1..0 %e A298884 ..0..0..1..1..1..1..0. .0..0..0..0..1..1..0. .0..1..1..1..1..1..1 %e A298884 ..0..1..0..1..1..1..1. .0..0..1..1..0..1..0. .1..0..1..1..1..0..0 %e A298884 ..0..0..1..1..0..0..0. .1..0..0..1..1..0..1. .1..0..1..1..1..0..1 %e A298884 ..1..1..0..0..1..1..1. .1..0..1..0..0..1..0. .1..0..1..0..0..1..0 %Y A298884 Cf. A298888. %K A298884 nonn,new %O A298884 1,2 %A A298884 _R. H. Hardin_, Jan 28 2018 %I A298854 %S A298854 1,1,1,2,3,2,6,11,11,6,24,50,61,50,24,120,274,379,379,274,120,720, %T A298854 1764,2668,3023,2668,1764,720,5040,13068,21160,26193,26193,21160, %U A298854 13068,5040,40320,109584,187388,248092,270961,248092,187388,109584,40320,362880,1026576,1836396,2565080,2995125,2995125,2565080,1836396,1026576,362880 %N A298854 Triangle read by rows. Characteristic polynomials of Jacobi coordinates. %C A298854 This is just a different normalization of A223256 and A223257. %F A298854 P(0)=1 and P(n) = n * (x + 1) * P(n - 1) - (n - 1)^2 * x * P(n - 2). %e A298854 For n = 3, the polynomial is 6*x^3 + 11*x^2 + 11*x + 6. %e A298854 The first few polynomials, as a table: %e A298854 [ 1], %e A298854 [ 1, 1], %e A298854 [ 2, 3, 2], %e A298854 [ 6, 11, 11, 6], %e A298854 [ 24, 50, 61, 50, 24], %e A298854 [120, 274, 379, 379, 274, 120] %o A298854 (Sage) %o A298854 @cached_function %o A298854 def poly(n): %o A298854 x = polygen(ZZ, 'x') %o A298854 if n < 0: %o A298854 return x.parent().zero() %o A298854 elif n == 0: %o A298854 return x.parent().one() %o A298854 else: %o A298854 return n * (x + 1) * poly(n - 1) - (n - 1)**2 * x * poly(n - 2) %o A298854 A298854_row = lambda n: list(poly(n)) %o A298854 for n in (0..7): print(A298854_row(n)) %Y A298854 Closely related to A223256 and A223257. %Y A298854 Row sums are A002720. %Y A298854 Leftmost and rightmost columns are A000142. %Y A298854 Alternating row sums are A177145. %K A298854 tabl,nonn,easy,new %O A298854 0,4 %A A298854 _F. Chapoton_, Jan 27 2018 %I A298510 %S A298510 1,-1,5,-15,-489,-2865,35685,-135135,-5897745,58437855,3061162125, %T A298510 -39296062575,-2278224696825,-33411730777425,-1496722493140875, %U A298510 -6190283353629375,-1563094742062478625,-17805713551427426625,1456700757237661060125,11729538718345143320625 %N A298510 a(n) = (n!)*2^(n-1)*mu_h(n) where mu_h is the hypergeometric Moebius function associated to the Dirichlet character modulo 4 h={1,0,-1,0,1,...} (see comment). %C A298510 Let h be a sequence and define the function g_h(x) = Sum_{1<=k<=1/x} h(k)/k*(k*x+1)/2 on the half-open interval on ]0,1]. Then the hypergeometric Moebius function associated to h is defined by the recursion Sum_{k=1..n} (mu_h(k)/k)*g_h(k/n) = 1/n. Here h(n)=1,0,-1,0,1,0,-1,... %C A298510 a(n) allows us to characterize primes and primes modulo 4: %C A298510 a(n)==0 (mod n) iff n is an odd composite number. %C A298510 a(n)==1 (mod n) iff n is a power of 2 or an odd prime of the form 4k+1. %C A298510 a(n)==n-1 (mod n) iff n is 1, 2, or an odd prime the of form 4k+3. %D A298510 B. Cloitre, The Riemann hypothesis is a topological property, in preparation (2018-....) %p A298510 a_list := proc(len) local s, gh, b: s := x -> select(k->modp(k,2)=1, [$1..1/x]); %p A298510 gh := x -> add((-1)^iquo(k,2)/k*(k*x+1), k in s(x)): b := proc(n) option remember; %p A298510 if n<=0 then 0 else 1/n - add(b(k)*gh(k/n), k=1..n-1)/2 fi end; %p A298510 seq(2^(k-1)*k!*b(k), k=1..len) end: a_list(20); # _Peter Luschny_, Jan 27 2018 %o A298510 (PARI) v=vector(1000); b(n)=if(n<0,0,v[n]); v[1]=1; gh(x)=sum(k=1,1/x, if(k%2,(-1)^(k\2),0)/k*(k*x+1)/2); for(n=2,#v, v[n]=1/n-sum(k=1,n-1,b(k)*gh(k/n))); a(n)=b(n)*n!*2^(n-1); %K A298510 sign,new %O A298510 1,3 %A A298510 _Benoit Cloitre_, Jan 20 2018 %I A297773 %S A297773 1,1,1,1,2,1,2,2,2,2,2,1,2,2,2,2,2,1,2,2,2,2,2,1,2,2,3,3,3,2,1,2,2,2, %T A297773 3,2,2,3,3,3,2,3,2,3,3,2,3,3,2,2,3,2,3,3,3,2,2,3,3,2,2,1,2,2,3,3,2,2, %U A297773 3,3,3,2,3,2,2,3,3,2,3,3,2,3,2,3,3,3 %N A297773 Number of distinct runs in base-5 digits of n. %C A297773 Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. %H A297773 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297773 8^8 in base 5: 1,3,2,4,3,3,3,2,3,3,1; eight runs, of which 6 are distinct, so that a(8^8) = 6. %t A297773 b = 5; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297773 Table[s[n], {n, 1, 200}] %Y A297773 Cf. A043557 (number of runs, not necessarily distinct), A297770, A043532. %K A297773 nonn,base,easy,new %O A297773 1,5 %A A297773 _Clark Kimberling_, Jan 27 2018 %I A297772 %S A297772 1,1,1,2,1,2,2,2,2,1,2,2,2,2,1,2,2,3,3,2,1,2,2,3,2,2,3,3,2,3,2,2,3,2, %T A297772 3,3,2,2,3,2,2,1,2,3,3,2,2,2,3,3,2,3,2,3,2,3,3,2,2,2,2,2,1,2,2,3,3,2, %U A297772 3,3,3,3,3,3,4,3,3,4,3,2,3,3,3,2,1,2 %N A297772 Number of distinct runs in base-4 digits of n. %C A297772 Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. %H A297772 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297772 123456 in base-4: 1,3,2,0,2,1,0,0,0; seven runs, of which 5 are distinct, so that a(123456) = 5. %t A297772 b = 4; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297772 Table[s[n], {n, 1, 200}] %Y A297772 Cf. A043556 (number of runs, not necessarily distinct), A297770. %K A297772 nonn,base,easy,new %O A297772 1,4 %A A297772 _Clark Kimberling_, Jan 27 2018 %I A297771 %S A297771 1,1,2,1,2,2,2,1,2,2,3,2,1,2,3,2,2,2,3,2,3,2,2,2,2,1,2,2,3,2,3,3,3,3, %T A297771 3,2,3,3,2,1,2,3,3,2,3,3,3,3,3,2,3,2,2,2,3,2,3,3,3,2,3,3,3,3,3,3,2,2, %U A297771 3,2,3,2,3,3,3,2,3,2,2,1,2,2,3,3,3,3 %N A297771 Number of distinct runs in base-3 digits of n. %C A297771 Every positive integers occurs infinitely many times. See A297770 for a guide to related sequences. %H A297771 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297771 1040 in base-3: 1,1,0,2,1,1,2; five runs, of which 3 are distinct, so that a(1040) = 3. %t A297771 b = 3; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297771 Table[s[n], {n, 1, 200}] %Y A297771 Cf. A043555 (number of runs, not necessarily distinct), A297770. %K A297771 nonn,base,easy,new %O A297771 1,3 %A A297771 _Clark Kimberling_, Jan 26 2018 %I A297770 %S A297770 1,2,1,2,2,2,1,2,2,2,3,2,3,2,1,2,2,3,3,3,2,3,3,2,3,3,2,2,3,2,1,2,2,3, %T A297770 3,2,3,4,3,3,3,2,3,4,3,3,3,2,3,4,2,4,3,2,3,2,3,3,3,2,3,2,1,2,2,3,3,3, %U A297770 3,4,3,3,2,3,4,3,4,4,3,3,3,3,4,3,2,3 %N A297770 Number of distinct runs in base-2 digits of n. %C A297770 Every positive integers occurs infinitely many times. %C A297770 *** %C A297770 Guide to related sequences: %C A297770 Base b # runs # distinct runs %C A297770 2 A005811 A297770 %C A297770 3 A043555 A297771 %C A297770 4 A043556 A297772 %C A297770 5 A043557 A297773 %C A297770 6 A043558 A297774 %C A297770 7 A043559 A297775 %C A297770 8 A043560 A297776 %C A297770 9 A043561 A297777 %C A297770 10 A043562 A297778 %C A297770 11 A043563 A297779 %C A297770 12 A043564 A297780 %C A297770 13 A043565 A297781 %C A297770 14 A043566 A297782 %C A297770 15 A043567 A297783 %C A297770 16 A043568 A297784 %H A297770 Clark Kimberling, Table of n, a(n) for n = 1..10000 %e A297770 27 in base-2: 1,1,0,1,1; three runs, of which 2 are distinct: 0 and 11, so that a(27) = 2. %t A297770 b = 2; s[n_] := Length[Union[Split[IntegerDigits[n, b]]]] %t A297770 Table[s[n], {n, 1, 200}] %Y A297770 Cf. A005811 (number of runs, not necessarily distinct). %K A297770 nonn,base,easy,new %O A297770 1,2 %A A297770 _Clark Kimberling_, Jan 26 2018 %I A298792 %S A298792 1,3,6,9,11,14,19,22,23,25,30 %N A298792 Coordination sequence for the bil tiling (or net) with respect to a trivalent node of the first type. %H A298792 Reticular Chemistry Structure Resource (RCSR), The bil tiling (or net) %Y A298792 Cf. A298793, A298794, A298795. %K A298792 nonn,more,new %O A298792 0,2 %A A298792 _N. J. A. Sloane_, Jan 27 2018 %I A298791 %S A298791 1,5,12,22,37,55,75,101,130,160,197,237,277,325,376,426,485,547,607, %T A298791 677,750,820,901,985,1065,1157,1252,1342,1445,1551,1651,1765,1882, %U A298791 1992,2117,2245,2365,2501,2640,2770,2917,3067,3207,3365,3526,3676,3845,4017,4177 %N A298791 Partial sums of A298789. %H A298791 Rémy Sigrist, Table of n, a(n) for n = 0..1000 %F A298791 Conjectures from _Colin Barker_, Jan 29 2018: (Start) %F A298791 G.f.: (1 + x)^4*(1 + x^2) / ((1 - x)^3*(1 + x + x^2)^2). %F A298791 a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n>6. %F A298791 (End) %Y A298791 Cf. A298789. %K A298791 nonn,new %O A298791 0,2 %A A298791 _N. J. A. Sloane_, Jan 27 2018 %E A298791 More terms from _Rémy Sigrist_, Jan 28 2018 %I A298790 %S A298790 1,4,11,23,36,53,77,100,127,163,196,233,281,324,371,431,484,541,613, %T A298790 676,743,827,900,977,1073,1156,1243,1351,1444,1541,1661,1764,1871, %U A298790 2003,2116,2233,2377,2500,2627,2783,2916,3053,3221,3364,3511,3691,3844,4001,4193 %N A298790 Partial sums of A298788. %H A298790 Rémy Sigrist, Table of n, a(n) for n = 0..1000 %F A298790 Conjectures from _Colin Barker_, Jan 29 2018: (Start) %F A298790 G.f.: (1 + x)^2*(1 + x + 4*x^2 + x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)^2). %F A298790 a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n>6. %F A298790 (End) %Y A298790 Cf. A298788. %K A298790 nonn,new %O A298790 0,2 %A A298790 _N. J. A. Sloane_, Jan 27 2018 %E A298790 More terms from _Rémy Sigrist_, Jan 28 2018 %I A298789 %S A298789 1,4,7,10,15,18,20,26,29,30,37,40,40,48,51,50,59,62,60,70,73,70,81,84, %T A298789 80,92,95,90,103,106,100,114,117,110,125,128,120,136,139,130,147,150, %U A298789 140,158,161,150,169,172,160,180,183,170,191,194,180,202,205,190 %N A298789 Coordination sequence for bey tiling (or net) with respect to a tetravalent node. %H A298789 Rémy Sigrist, Table of n, a(n) for n = 0..1000 %H A298789 Reticular Chemistry Structure Resource (RCSR), The bey tiling (or net) %H A298789 Rémy Sigrist, Illustration of the first terms %H A298789 Rémy Sigrist, PARI program for A298789 %F A298789 Conjectures from _Colin Barker_, Jan 29 2018: (Start) %F A298789 G.f.: (1 + x)^4*(1 + x^2) / ((1 - x)^2*(1 + x + x^2)^2). %F A298789 a(n) = 2*a(n-3) - a(n-6) for n>6. %F A298789 (End) %o A298789 (PARI) See Links section. %Y A298789 Cf. A298788, A298790, A298791. %K A298789 nonn,new %O A298789 0,2 %A A298789 _N. J. A. Sloane_, Jan 27 2018 %E A298789 More terms from _Rémy Sigrist_, Jan 28 2018 %I A298788 %S A298788 1,3,7,12,13,17,24,23,27,36,33,37,48,43,47,60,53,57,72,63,67,84,73,77, %T A298788 96,83,87,108,93,97,120,103,107,132,113,117,144,123,127,156,133,137, %U A298788 168,143,147,180,153,157,192,163,167,204,173,177,216,183,187,228 %N A298788 Coordination sequence for bey tiling (or net) with respect to a trivalent node. %H A298788 Rémy Sigrist, Table of n, a(n) for n = 0..1000 %H A298788 Reticular Chemistry Structure Resource (RCSR), The bey tiling (or net) %H A298788 Rémy Sigrist, Illustration of the first terms %H A298788 Rémy Sigrist, PARI program for A298788 %F A298788 Conjectures from _Colin Barker_, Jan 29 2018: (Start) %F A298788 G.f.: (1 + x)^2*(1 + x + 4*x^2 + x^3 + x^4) / ((1 - x)^2*(1 + x + x^2)^2). %F A298788 a(n) = 2*a(n-3) - a(n-6) for n>6. %F A298788 (End) %o A298788 (PARI) See Links section. %Y A298788 Cf. A298789, A298790, A298791. %K A298788 nonn,new %O A298788 0,2 %A A298788 _N. J. A. Sloane_, Jan 27 2018 %E A298788 More terms from _Rémy Sigrist_, Jan 28 2018 %I A298639 %S A298639 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,21,22,23,24,25,26, %T A298639 27,30,31,32,33,34,35,36,40,41,42,43,44,45,50,51,52,53,54,60,61,62,63, %U A298639 70,71,72,80,81,90,100,101,102,103,104,105,106,107,108,110,111,112,113,114 %N A298639 Numbers k such that the digital sum of k and the digital root of k have the same parity. %C A298639 Numbers k such that A113217(k) = A179081(k). %C A298639 Complement of A298638. %C A298639 Agrees with A039691 until a(65): A039691(65) = 109 is not in this sequence. %H A298639 J. Stauduhar, Table of n, a(n) for n = 1..10000 %t A298639 fQ[n_] := Mod[Plus @@ IntegerDigits@n, 2] == Mod[Mod[n -1, 9] +1, 2]; fQ[0] = True; Select[ Range[0, 104], fQ] (* _Robert G. Wilson v_, Jan 26 2018 *) %o A298639 (PYTHON) %o A298639 #Digital sum of n. %o A298639 def ds(n): %o A298639 if n < 10: %o A298639 return n %o A298639 return n % 10 + ds(n//10) %o A298639 def A298639(term_count): %o A298639 seq = [] %o A298639 m = 0 %o A298639 n = 1 %o A298639 while n <= term_count: %o A298639 s = ds(m) %o A298639 r = ((m - 1) % 9) + 1 if m else 0 %o A298639 if s % 2 == r % 2: %o A298639 seq.append(m) %o A298639 n += 1 %o A298639 m += 1 %o A298639 return seq %o A298639 print(A298639(100)) %o A298639 (PARI) dr(n)=if(n, (n-1)%9+1); %o A298639 isok(n) = (sumdigits(n) % 2) == (dr(n) % 2); \\ _Michel Marcus_, Jan 26 2018 %o A298639 (PARI) is(n)=bittest(sumdigits(n)-(n-1)%9,0)||!n \\ _M. F. Hasler_, Jan 26 2018 %Y A298639 Cf. A007953, A010888, A113217, A179081, A298638, A039691. %K A298639 nonn,easy,base,new %O A298639 1,3 %A A298639 _J. Stauduhar_, Jan 26 2018 %I A298846 %S A298846 0,1,1,1,4,1,2,17,17,2,3,49,48,49,3,5,166,146,146,166,5,8,573,424,466, %T A298846 424,573,8,13,1933,1274,1446,1446,1274,1933,13,21,6538,3820,4648,5101, %U A298846 4648,3820,6538,21,34,22165,11529,14888,18191,18191,14888,11529,22165,34 %N A298846 T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 3, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298846 Table starts %C A298846 ..0.....1.....1......2......3.......5........8........13.........21.........34 %C A298846 ..1.....4....17.....49....166.....573.....1933......6538......22165......75089 %C A298846 ..1....17....48....146....424....1274.....3820.....11529......34783.....104826 %C A298846 ..2....49...146....466...1446....4648....14888.....47399.....150849.....480015 %C A298846 ..3...166...424...1446...5101...18191....62393....214017.....735090....2521128 %C A298846 ..5...573..1274...4648..18191...74685...290474...1134326....4476521...17504192 %C A298846 ..8..1933..3820..14888..62393..290474..1276034...5745665...26250580..117940632 %C A298846 .13..6538.11529..47399.214017.1134326..5745665..30242236..163316554..862848634 %C A298846 .21.22165.34783.150849.735090.4476521.26250580.163316554.1056423010.6667708149 %H A298846 R. H. Hardin, Table of n, a(n) for n = 1..337 %F A298846 Empirical for column k: %F A298846 k=1: a(n) = a(n-1) +a(n-2) %F A298846 k=2: a(n) = 3*a(n-1) +a(n-2) +2*a(n-3) -2*a(n-4) -4*a(n-5) for n>6 %F A298846 k=3: [order 11] for n>13 %F A298846 k=4: [order 24] for n>27 %e A298846 Some solutions for n=6 k=6 %e A298846 ..0..1..1..1..0..0. .0..0..1..1..1..0. .0..1..0..1..0..1. .0..0..0..1..0..1 %e A298846 ..1..0..1..0..1..0. .0..1..0..1..0..1. .1..0..1..1..1..0. .0..1..1..1..1..0 %e A298846 ..0..1..1..1..0..1. .1..0..1..1..1..1. .1..1..1..1..1..1. .0..1..1..1..1..1 %e A298846 ..1..1..1..1..1..1. .1..1..1..1..1..0. .1..0..1..1..1..0. .1..1..1..1..1..0 %e A298846 ..0..1..1..1..0..1. .1..0..1..1..1..0. .0..1..0..1..0..1. .0..1..1..1..1..0 %e A298846 ..1..0..1..0..1..0. .0..1..1..0..0..0. .0..0..1..1..1..0. .1..0..1..0..0..0 %Y A298846 Column 1 is A000045(n-1). %Y A298846 Column 2 is A297817. %Y A298846 Column 3 is A297988. %Y A298846 Column 4 is A297989. %K A298846 nonn,tabl,new %O A298846 1,5 %A A298846 _R. H. Hardin_, Jan 27 2018 %I A298845 %S A298845 8,1933,3820,14888,62393,290474,1276034,5745665,26250580,117940632, %T A298845 529830532,2398014801,10851357559,49048617886,221887809644, %U A298845 1004033294806,4542287520456,20553927069804,93029936095342,421086056398025 %N A298845 Number of nX7 0..1 arrays with every element equal to 1, 2, 3, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298845 Column 7 of A298846. %H A298845 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298845 Some solutions for n=5 %e A298845 ..0..1..0..1..0..1..0. .0..0..1..0..1..0..1. .0..1..0..1..0..1..1 %e A298845 ..1..0..1..1..1..0..1. .0..1..0..1..1..1..0. .1..0..0..0..1..0..1 %e A298845 ..1..1..1..1..1..1..1. .0..1..1..1..1..1..1. .0..0..0..0..0..0..1 %e A298845 ..1..0..1..1..1..0..1. .0..1..0..1..1..1..0. .1..0..0..0..1..0..1 %e A298845 ..0..1..0..1..0..1..0. .0..0..1..0..1..0..1. .0..1..0..1..0..1..1 %Y A298845 Cf. A298846. %K A298845 nonn,new %O A298845 1,1 %A A298845 _R. H. Hardin_, Jan 27 2018 %I A298844 %S A298844 5,573,1274,4648,18191,74685,290474,1134326,4476521,17504192,68460956, %T A298844 268946027,1056908182,4152467108,16322047853,64182331628,252466069300, %U A298844 993456216706,3910523487359,15397203073949,60640330684857 %N A298844 Number of nX6 0..1 arrays with every element equal to 1, 2, 3, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298844 Column 6 of A298846. %H A298844 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298844 Some solutions for n=6 %e A298844 ..0..1..1..0..0..0. .0..0..0..1..0..1. .0..1..0..1..0..1. .0..0..1..1..1..0 %e A298844 ..1..0..1..1..1..0. .0..1..1..1..1..0. .1..0..0..0..1..0. .0..1..0..1..0..1 %e A298844 ..1..1..1..1..1..0. .0..1..1..1..1..1. .0..0..0..0..0..0. .1..0..1..1..1..0 %e A298844 ..1..0..1..1..1..1. .1..1..1..1..1..0. .1..0..0..0..1..0. .1..1..1..1..1..1 %e A298844 ..0..1..0..1..0..1. .0..1..1..1..1..0. .0..1..0..1..0..1. .1..0..1..1..1..0 %e A298844 ..0..0..1..1..1..0. .1..0..1..0..0..0. .1..0..0..0..1..1. .0..1..0..1..0..1 %Y A298844 Cf. A298846. %K A298844 nonn,new %O A298844 1,1 %A A298844 _R. H. Hardin_, Jan 27 2018 %I A298843 %S A298843 3,166,424,1446,5101,18191,62393,214017,735090,2521128,8674496, %T A298843 29900417,103081768,355521461,1226654477,4233860529,14618644242, %U A298843 50489092706,174414948926,602632498696,2082528838007,7197611462672,24879153769164 %N A298843 Number of nX5 0..1 arrays with every element equal to 1, 2, 3, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298843 Column 5 of A298846. %H A298843 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298843 Some solutions for n=7 %e A298843 ..0..0..0..0..0. .0..1..0..1..0. .0..1..1..1..0 %e A298843 ..0..1..1..1..0. .1..0..0..0..1. .1..0..1..0..1 %e A298843 ..1..0..1..0..1. .0..0..0..0..0. .0..1..1..1..0 %e A298843 ..0..1..1..1..0. .1..0..0..0..1. .1..1..1..1..1 %e A298843 ..1..1..1..1..1. .0..1..0..1..0. .0..1..1..1..0 %e A298843 ..0..1..1..1..0. .1..0..0..0..1. .1..0..1..0..1 %e A298843 ..1..0..1..0..1. .1..1..1..1..1. .0..1..1..1..0 %Y A298843 Cf. A298846. %K A298843 nonn,new %O A298843 1,1 %A A298843 _R. H. Hardin_, Jan 27 2018 %I A298842 %S A298842 0,4,48,466,5101,74685,1276034,30242236,1056423010,50039513246, %T A298842 3406811335705,353276120560238,52792202092264104 %N A298842 Number of nXn 0..1 arrays with every element equal to 1, 2, 3, 6 or 8 king-move adjacent elements, with upper left element zero. %C A298842 Diagonal of A298846. %e A298842 Some solutions for n=6 %e A298842 ..0..1..0..1..0..1. .0..1..0..1..0..1. .0..0..0..1..0..1. .0..1..1..1..0..0 %e A298842 ..1..0..1..1..1..0. .1..0..0..0..1..0. .0..1..1..1..1..0. .1..0..1..0..1..0 %e A298842 ..1..1..1..1..1..1. .0..0..0..0..0..0. .0..1..1..1..1..1. .1..1..1..1..0..1 %e A298842 ..1..0..1..1..1..0. .1..0..0..0..1..0. .1..1..1..1..1..0. .0..1..1..1..1..1 %e A298842 ..0..1..0..1..0..1. .0..1..0..1..0..1. .0..1..1..1..1..0. .0..1..1..1..0..1 %e A298842 ..0..0..1..1..1..0. .1..0..0..0..1..1. .1..0..1..0..0..0. .0..0..0..1..1..0 %Y A298842 Cf. A298846. %K A298842 nonn,new %O A298842 1,2 %A A298842 _R. H. Hardin_, Jan 27 2018 %I A298841 %S A298841 0,0,0,0,1,0,0,3,3,0,0,6,9,6,0,0,17,21,21,17,0,0,41,127,110,127,41,0, %T A298841 0,104,513,1045,1045,513,104,0,0,261,2440,7322,18075,7322,2440,261,0, %U A298841 0,655,11458,60497,242872,242872,60497,11458,655,0,0,1646,53727,482261,3570731 %N A298841 T(n,k)=Number of nXk 0..1 arrays with every element equal to 2, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298841 Table starts %C A298841 .0...0.....0.......0.........0............0..............0................0 %C A298841 .0...1.....3.......6........17...........41............104..............261 %C A298841 .0...3.....9......21.......127..........513...........2440............11458 %C A298841 .0...6....21.....110......1045.........7322..........60497...........482261 %C A298841 .0..17...127....1045.....18075.......242872........3570731.........52303834 %C A298841 .0..41...513....7322....242872......5918418......160441457.......4287985074 %C A298841 .0.104..2440...60497...3570731....160441457.....7844725524.....380985444657 %C A298841 .0.261.11458..482261..52303834...4287985074...380985444657...33608679844048 %C A298841 .0.655.53727.3886764.766622466.114829835558.18510417612668.2967985050748586 %H A298841 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A298841 Empirical for column k: %F A298841 k=1: a(n) = a(n-1) %F A298841 k=2: a(n) = a(n-1) +3*a(n-2) +2*a(n-3) %F A298841 k=3: [order 12] for n>14 %F A298841 k=4: [order 45] for n>48 %e A298841 Some solutions for n=5 k=5 %e A298841 ..0..0..0..1..1. .0..0..1..1..1. .0..0..0..1..1. .0..0..0..0..0 %e A298841 ..0..0..0..1..1. .0..0..1..1..1. .0..0..0..1..1. .0..1..1..1..0 %e A298841 ..0..0..0..1..1. .0..0..1..1..1. .0..0..0..1..1. .0..1..1..1..0 %e A298841 ..1..1..1..1..0. .0..1..0..0..0. .1..1..1..0..0. .0..1..1..1..0 %e A298841 ..1..1..1..0..0. .1..1..0..0..0. .1..1..1..1..0. .0..0..0..0..0 %Y A298841 Column 2 is A297972. %K A298841 nonn,tabl,new %O A298841 1,8 %A A298841 _R. H. Hardin_, Jan 27 2018 %I A298840 %S A298840 0,104,2440,60497,3570731,160441457,7844725524,380985444657, %T A298840 18510417612668,902275957883352,43936865148320525,2140933146128529606, %U A298840 104313951368150470461,5082784700709399173199,247666770498486555848961 %N A298840 Number of nX7 0..1 arrays with every element equal to 2, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298840 Column 7 of A298841. %H A298840 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298840 Some solutions for n=4 %e A298840 ..0..0..0..0..1..1..1. .0..0..0..0..1..1..1. .0..0..1..1..1..0..0 %e A298840 ..0..1..1..0..1..1..1. .0..0..0..0..1..1..1. .0..0..1..1..1..0..0 %e A298840 ..1..1..0..0..1..1..1. .0..0..0..0..1..1..1. .0..0..1..1..1..0..0 %e A298840 ..1..0..0..0..1..1..1. .0..0..0..0..1..1..1. .0..0..1..1..1..0..0 %Y A298840 Cf. A298841. %K A298840 nonn,new %O A298840 1,2 %A A298840 _R. H. Hardin_, Jan 27 2018 %I A298839 %S A298839 0,41,513,7322,242872,5918418,160441457,4287985074,114829835558, %T A298839 3085289326902,82801989921924,2223892344859142,59722476102921078, %U A298839 1603938095913408003,43076775918779371087,1156907815094072294324 %N A298839 Number of nX6 0..1 arrays with every element equal to 2, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298839 Column 6 of A298841. %H A298839 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298839 Some solutions for n=4 %e A298839 ..0..1..1..0..0..0. .0..0..0..1..0..0. .0..0..0..1..1..1. .0..0..0..0..0..0 %e A298839 ..0..0..1..0..0..0. .0..0..0..1..1..0. .0..1..0..1..1..1. .0..0..0..0..0..0 %e A298839 ..0..1..1..0..0..0. .0..0..0..1..1..0. .1..1..0..1..1..1. .0..0..0..0..0..0 %e A298839 ..0..0..1..0..0..0. .0..0..0..1..0..0. .1..0..0..1..1..1. .0..0..0..0..0..0 %Y A298839 Cf. A298841. %K A298839 nonn,new %O A298839 1,2 %A A298839 _R. H. Hardin_, Jan 27 2018 %I A298838 %S A298838 0,17,127,1045,18075,242872,3570731,52303834,766622466,11271546283, %T A298838 165665212660,2435912683020,35819238408346,526724402976581, %U A298838 7745695689623062,113904017404740204,1675017069147795397 %N A298838 Number of nX5 0..1 arrays with every element equal to 2, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298838 Column 5 of A298841. %H A298838 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298838 Some solutions for n=5 %e A298838 ..0..0..1..1..1. .0..0..1..1..1. .0..0..0..1..1. .0..0..1..1..1 %e A298838 ..0..0..1..1..1. .0..0..1..1..1. .0..0..0..1..1. .0..0..1..1..1 %e A298838 ..1..0..1..1..1. .0..1..0..0..0. .0..0..0..1..1. .0..0..1..1..1 %e A298838 ..1..1..0..0..0. .1..1..0..0..0. .1..1..1..1..0. .0..1..0..0..0 %e A298838 ..1..0..0..0..0. .1..1..0..0..0. .1..1..1..0..0. .1..1..0..0..0 %Y A298838 Cf. A298841. %K A298838 nonn,new %O A298838 1,2 %A A298838 _R. H. Hardin_, Jan 27 2018 %I A298837 %S A298837 0,6,21,110,1045,7322,60497,482261,3886764,31419679,253669718, %T A298837 2050750767,16574522884,133980763315,1083047418757,8754987461582, %U A298837 70773060189635,572110944739376,4624804772439213,37385796035240692,302217747405919419 %N A298837 Number of nX4 0..1 arrays with every element equal to 2, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298837 Column 4 of A298841. %H A298837 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298837 Empirical: a(n) = 9*a(n-1) +17*a(n-2) -195*a(n-3) -301*a(n-4) +2160*a(n-5) +2724*a(n-6) -12908*a(n-7) -15160*a(n-8) +47906*a(n-9) +54649*a(n-10) -121788*a(n-11) -119168*a(n-12) +219680*a(n-13) +109795*a(n-14) -358602*a(n-15) +332162*a(n-16) +436116*a(n-17) -1047261*a(n-18) -907623*a(n-19) +1904805*a(n-20) +943907*a(n-21) -401499*a(n-22) -3080376*a(n-23) +756838*a(n-24) +2344863*a(n-25) -422428*a(n-26) -154363*a(n-27) -1739231*a(n-28) +1061307*a(n-29) +880883*a(n-30) -496873*a(n-31) -39576*a(n-32) -190649*a(n-33) +102350*a(n-34) -69812*a(n-35) -105271*a(n-36) +28788*a(n-37) +64379*a(n-38) +25435*a(n-39) -45832*a(n-40) -6076*a(n-41) +16110*a(n-42) +1038*a(n-43) -3416*a(n-44) +408*a(n-45) for n>48 %e A298837 Some solutions for n=7 %e A298837 ..0..0..1..1. .0..0..1..1. .0..0..0..0. .0..0..0..0. .0..0..1..1 %e A298837 ..0..1..1..0. .1..0..0..1. .0..0..0..0. .0..1..0..0. .0..1..1..1 %e A298837 ..0..0..0..0. .1..1..0..1. .0..0..0..0. .1..0..1..1. .1..0..0..0 %e A298837 ..1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1. .1..0..0..0 %e A298837 ..1..1..1..1. .0..0..0..0. .1..0..1..1. .0..0..0..0. .1..0..0..0 %e A298837 ..1..1..1..1. .0..0..0..0. .0..0..0..1. .0..0..0..0. .0..1..1..1 %e A298837 ..1..1..1..1. .0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..1..1 %Y A298837 Cf. A298841. %K A298837 nonn,new %O A298837 1,2 %A A298837 _R. H. Hardin_, Jan 27 2018 %I A298836 %S A298836 0,3,9,21,127,513,2440,11458,53727,254534,1202650,5692243,26943691, %T A298836 127540324,603796896,2858444154,13532486972,64065914939,303303284790, %U A298836 1435912281177,6797962755503,32183241315912,152363457753491 %N A298836 Number of nX3 0..1 arrays with every element equal to 2, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298836 Column 3 of A298841. %H A298836 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298836 Empirical: a(n) = 5*a(n-1) +6*a(n-2) -31*a(n-3) -40*a(n-4) +100*a(n-5) +105*a(n-6) -167*a(n-7) -101*a(n-8) +118*a(n-9) +29*a(n-10) -19*a(n-11) +4*a(n-12) for n>14 %e A298836 Some solutions for n=7 %e A298836 ..0..0..0. .0..0..0. .0..1..1. .0..0..0. .0..0..0. .0..0..0. .0..0..0 %e A298836 ..0..0..0. .0..0..0. .0..0..1. .0..0..0. .0..0..0. .0..1..0. .0..0..0 %e A298836 ..0..0..0. .0..0..0. .1..0..1. .0..0..0. .1..1..1. .1..1..1. .0..0..0 %e A298836 ..1..1..1. .0..0..0. .1..1..1. .1..1..1. .1..1..1. .0..0..0. .1..1..1 %e A298836 ..1..0..1. .0..0..0. .0..0..0. .1..1..0. .1..1..1. .0..0..0. .0..1..1 %e A298836 ..0..1..0. .1..1..1. .0..0..0. .1..0..0. .1..1..1. .0..0..0. .0..0..1 %e A298836 ..0..0..0. .1..1..1. .0..0..0. .1..1..0. .1..1..1. .0..0..0. .0..0..0 %Y A298836 Cf. A298841. %K A298836 nonn,new %O A298836 1,2 %A A298836 _R. H. Hardin_, Jan 27 2018 %I A298835 %S A298835 0,1,9,110,18075,5918418,7844725524,33608679844048,476321271854010112 %N A298835 Number of nXn 0..1 arrays with every element equal to 2, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298835 Diagonal of A298841. %e A298835 Some solutions for n=5 %e A298835 ..0..0..1..1..1. .0..0..0..0..0. .0..0..0..1..1. .0..0..0..1..1 %e A298835 ..0..0..1..1..1. .0..0..0..0..0. .0..0..0..1..0. .0..0..0..1..1 %e A298835 ..1..0..1..1..1. .0..0..0..0..0. .1..1..1..0..0. .0..0..0..1..1 %e A298835 ..1..1..0..0..0. .0..0..0..0..0. .1..1..1..0..0. .1..1..1..0..1 %e A298835 ..1..0..0..0..0. .0..0..0..0..0. .1..1..1..0..0. .1..1..1..0..0 %Y A298835 Cf. A298841. %K A298835 nonn,new %O A298835 1,3 %A A298835 _R. H. Hardin_, Jan 27 2018 %I A298834 %S A298834 0,1,1,0,4,0,1,4,4,1,0,16,1,16,0,1,48,6,6,48,1,0,88,10,68,10,88,0,1, %T A298834 240,15,141,141,15,240,1,0,704,60,489,1590,489,60,704,0,1,1600,128, %U A298834 2774,3282,3282,2774,128,1600,1,0,4032,267,9849,27915,16551,27915,9849,267,4032,0 %N A298834 T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298834 Table starts %C A298834 .0....1...0.....1.......0........1.........0...........1.............0 %C A298834 .1....4...4....16......48.......88.......240.........704..........1600 %C A298834 .0....4...1.....6......10.......15........60.........128...........267 %C A298834 .1...16...6....68.....141......489......2774........9849.........39101 %C A298834 .0...48..10...141....1590.....3282.....27915......246700.......1055145 %C A298834 .1...88..15...489....3282....16551....226883.....1917869......16022685 %C A298834 .0..240..60..2774...27915...226883...5097700....70784399.....995832826 %C A298834 .1..704.128..9849..246700..1917869..70784399..2031564141...35098724239 %C A298834 .0.1600.267.39101.1055145.16022685.995832826.35098724239.1185589525594 %H A298834 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A298834 Empirical for column k: %F A298834 k=1: a(n) = a(n-2) %F A298834 k=2: a(n) = 2*a(n-1) +8*a(n-3) -8*a(n-4) -8*a(n-5) %F A298834 k=3: [order 19] for n>20 %F A298834 k=4: [order 63] for n>65 %e A298834 Some solutions for n=5 k=4 %e A298834 ..0..0..0..1. .0..0..0..0. .0..0..1..1. .0..1..0..1. .0..0..1..0 %e A298834 ..0..0..0..1. .1..1..0..0. .0..0..0..0. .0..1..0..1. .0..0..0..1 %e A298834 ..1..1..1..1. .1..1..1..1. .1..1..1..1. .0..0..0..0. .1..1..1..1 %e A298834 ..0..0..1..0. .0..1..0..0. .1..1..1..1. .1..0..1..0. .0..0..1..1 %e A298834 ..0..0..1..0. .0..1..0..0. .1..1..1..1. .1..0..1..0. .0..0..0..0 %Y A298834 Column 2 is A298448. %K A298834 nonn,tabl,new %O A298834 1,5 %A A298834 _R. H. Hardin_, Jan 27 2018 %I A298833 %S A298833 0,240,60,2774,27915,226883,5097700,70784399,995832826,17401811425, %T A298833 288129465222,4336914140747,73188995403769,1218049349765186, %U A298833 19154346227374288,317120961246926128,5242138886321970318 %N A298833 Number of nX7 0..1 arrays with every element equal to 1, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298833 Column 7 of A298834. %H A298833 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298833 Some solutions for n=5 %e A298833 ..0..0..1..1..1..0..1. .0..0..1..1..0..0..0. .0..0..0..0..1..1..1 %e A298833 ..1..1..1..1..1..1..0. .0..0..1..1..0..0..0. .0..0..0..0..1..1..1 %e A298833 ..1..1..0..0..0..1..1. .1..1..0..1..0..0..0. .1..1..1..1..0..0..0 %e A298833 ..0..1..0..0..0..1..0. .0..0..1..0..1..1..1. .1..1..1..1..0..1..1 %e A298833 ..0..1..0..0..0..1..0. .0..0..0..1..1..1..1. .1..1..1..1..0..1..1 %Y A298833 Cf. A298834. %K A298833 nonn,new %O A298833 1,2 %A A298833 _R. H. Hardin_, Jan 27 2018 %I A298832 %S A298832 1,88,15,489,3282,16551,226883,1917869,16022685,179166054,1784881081, %T A298832 15874121652,167851452385,1695259719084,15893552680555, %U A298832 162450100990858,1634524710070620,15778446550779276,158567398271907840 %N A298832 Number of nX6 0..1 arrays with every element equal to 1, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298832 Column 6 of A298834. %H A298832 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298832 Some solutions for n=5 %e A298832 ..0..0..0..1..1..0. .0..0..0..0..1..1. .0..0..0..1..1..0. .0..0..1..0..1..1 %e A298832 ..0..0..0..1..1..0. .1..1..0..0..0..0. .0..0..0..1..1..0. .0..0..0..1..1..1 %e A298832 ..0..0..0..1..0..0. .1..1..0..1..1..1. .0..0..0..1..0..0. .1..1..1..0..0..0 %e A298832 ..1..1..1..1..0..0. .1..0..0..1..1..1. .1..1..1..1..0..0. .1..1..1..0..0..0 %e A298832 ..0..0..1..1..1..1. .1..0..0..1..1..1. .0..0..1..1..0..0. .1..1..1..0..0..0 %Y A298832 Cf. A298834. %K A298832 nonn,new %O A298832 1,2 %A A298832 _R. H. Hardin_, Jan 27 2018 %I A298831 %S A298831 0,48,10,141,1590,3282,27915,246700,1055145,7129209,56638725, %T A298831 310179586,2031847146,14612093372,88989021585,580066345101, %U A298831 3961760881902,25211203481003,164396604060820,1096627513193338,7101386707723193 %N A298831 Number of nX5 0..1 arrays with every element equal to 1, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298831 Column 5 of A298834. %H A298831 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298831 Some solutions for n=5 %e A298831 ..0..0..1..1..1. .0..0..1..1..1. .0..0..1..0..1. .0..0..1..1..0 %e A298831 ..0..0..1..1..1. .0..0..1..1..1. .1..1..1..1..0. .1..1..1..1..0 %e A298831 ..0..0..1..1..1. .0..0..1..1..1. .0..0..0..1..1. .0..0..0..1..1 %e A298831 ..1..0..0..0..0. .0..0..1..1..1. .0..0..0..1..1. .0..0..0..1..0 %e A298831 ..0..1..0..1..1. .0..0..1..1..1. .0..0..0..1..1. .0..0..0..1..0 %Y A298831 Cf. A298834. %K A298831 nonn,new %O A298831 1,2 %A A298831 _R. H. Hardin_, Jan 27 2018 %I A298830 %S A298830 1,16,6,68,141,489,2774,9849,39101,183074,768268,3094425,13600466, %T A298830 58825790,243388690,1044751712,4510287691,18993143352,80911731590, %U A298830 347373816744,1474949726778,6275196854211,26837387067655,114307141604206 %N A298830 Number of nX4 0..1 arrays with every element equal to 1, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298830 Column 4 of A298834. %H A298830 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298830 Empirical: a(n) = 4*a(n-1) +a(n-2) +49*a(n-3) -190*a(n-4) -142*a(n-5) -490*a(n-6) +3529*a(n-7) +931*a(n-8) -2445*a(n-9) -22916*a(n-10) +8373*a(n-11) +46085*a(n-12) +28964*a(n-13) -74055*a(n-14) -166635*a(n-15) +170052*a(n-16) +116703*a(n-17) +83747*a(n-18) -359546*a(n-19) +148326*a(n-20) +463402*a(n-21) -380831*a(n-22) -522218*a(n-23) +118848*a(n-24) +606224*a(n-25) +51024*a(n-26) -315707*a(n-27) -51586*a(n-28) -479161*a(n-29) -4310630*a(n-30) +6488972*a(n-31) -360817*a(n-32) +1226847*a(n-33) +97998*a(n-34) -2001512*a(n-35) -2307*a(n-36) -2820274*a(n-37) +1066042*a(n-38) +1038066*a(n-39) -416347*a(n-40) -1837978*a(n-41) +2558113*a(n-42) -327253*a(n-43) -301655*a(n-44) -874393*a(n-45) +2898299*a(n-46) +1530562*a(n-47) -1350912*a(n-48) -2229129*a(n-49) -648926*a(n-50) +481347*a(n-51) +250281*a(n-52) +86759*a(n-53) +184992*a(n-54) +330267*a(n-55) +26616*a(n-56) -139260*a(n-57) -132100*a(n-58) -965*a(n-59) +23028*a(n-60) +14606*a(n-61) -1538*a(n-62) -504*a(n-63) for n>65 %e A298830 Some solutions for n=5 %e A298830 ..0..0..1..1. .0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..0 %e A298830 ..0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..0 %e A298830 ..1..1..1..1. .1..1..1..1. .0..0..0..0. .0..0..0..0. .0..0..0..0 %e A298830 ..1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1 %e A298830 ..1..1..1..1. .1..1..1..1. .1..1..0..0. .0..0..1..1. .1..1..1..1 %Y A298830 Cf. A298834. %K A298830 nonn,new %O A298830 1,2 %A A298830 _R. H. Hardin_, Jan 27 2018 %I A298829 %S A298829 0,4,1,6,10,15,60,128,267,810,1878,4579,12408,30552,77105,200876, %T A298829 505862,1288511,3311514,8411436,21467751,54921524,139947224,357287611, %U A298829 912689144,2328342518,5944207749,15177139978,38734201690,98882999903,252436571832 %N A298829 Number of nX3 0..1 arrays with every element equal to 1, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298829 Column 3 of A298834. %H A298829 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298829 Empirical: a(n) = 2*a(n-1) +3*a(n-2) +5*a(n-3) -22*a(n-4) -15*a(n-5) +17*a(n-6) +44*a(n-7) -13*a(n-8) -49*a(n-9) +74*a(n-10) +38*a(n-11) -91*a(n-12) -61*a(n-13) +42*a(n-14) +24*a(n-15) -4*a(n-16) +a(n-17) -a(n-18) -a(n-19) for n>20 %e A298829 Some solutions for n=5 %e A298829 ..0..0..1. .0..0..1. .0..0..0. .0..1..0. .0..1..1. .0..0..0. .0..1..0 %e A298829 ..0..0..1. .0..0..1. .0..0..0. .0..1..0. .0..1..1. .0..0..0. .0..1..0 %e A298829 ..0..1..1. .1..1..1. .0..0..0. .1..1..1. .0..0..0. .1..1..1. .0..0..0 %e A298829 ..0..0..1. .0..0..1. .1..1..1. .1..0..1. .0..1..1. .1..1..1. .1..0..1 %e A298829 ..0..0..1. .0..0..1. .1..1..1. .1..0..1. .0..1..1. .1..1..1. .1..0..1 %Y A298829 Cf. A298834. %K A298829 nonn,new %O A298829 1,2 %A A298829 _R. H. Hardin_, Jan 27 2018 %I A298828 %S A298828 0,4,1,68,1590,16551,5097700,2031564141,1185589525594 %N A298828 Number of nXn 0..1 arrays with every element equal to 1, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298828 Diagonal of A298834. %e A298828 Some solutions for n=5 %e A298828 ..0..0..1..0..0. .0..0..0..1..0. .0..1..0..0..0. .0..0..1..1..1 %e A298828 ..1..1..1..1..1. .0..0..0..1..0. .0..1..0..0..0. .0..0..1..1..1 %e A298828 ..1..1..0..0..0. .0..0..0..1..1. .1..1..0..0..0. .0..0..1..1..1 %e A298828 ..0..1..0..0..0. .0..0..0..1..1. .0..1..1..1..1. .1..0..0..0..0 %e A298828 ..0..1..0..0..0. .0..0..0..1..1. .1..0..1..0..0. .0..1..0..1..1 %Y A298828 Cf. A298834. %K A298828 nonn,new %O A298828 1,2 %A A298828 _R. H. Hardin_, Jan 27 2018 %I A298704 %S A298704 10942177,33612487,38370391,350212139,431472421,594652609,616335793, %T A298704 795968851,867683191,885354367,982828577,1058353099,1278213241, %U A298704 1367427883,1966024733,2329521127,2734073753,2959246399,3067739581 %N A298704 Numbers that are the smallest of four consecutive primes, no two of which, along with the product of the other two, sum to a nonprime. %H A298704 Hans Havermann, Table of n, a(n) for n = 1..30 %e A298704 10942177, 10942187, 10942201, 10942207 are four consecutive primes. The six ways of adding the product of two of these to the sum of the other two yields 119731368805507, 119731521995971, 119731587649027, 119731631417971, 119731697071087, 119731850261971, all of which are prime. So 10942177 is a term of the sequence. %p A298704 with(numtheory): with(combinat): %p A298704 P:=proc(q) local a,b,c,d,k,n,ok,x,y; %p A298704 a:=0; b:=10942177; c:=10942187; d:=10942201; %p A298704 for n from 1 to q do ok:=1; a:=b; b:=c; c:=d; %p A298704 d:=nextprime(d); x:={a,b,c,d}; y:=choose(x,2); %p A298704 for k from 1 to 6 do if not isprime(convert(y[k],`*`)+convert(x minus y[k],`+`)) then ok:=0; break; fi; od; if ok=1 then print(a); fi; od; end: P(10^20); # _Paolo P. Lava_, Jan 25 2018 %K A298704 nonn,new %O A298704 1,1 %A A298704 _Hans Havermann_, Jan 24 2018 %I A298763 %S A298763 19,29,1303,3119,4933,6353,7841,10859,13933,24749,26513,28603,31069, %T A298763 33487,38609,43067,52387,53731,61979,78031,91781,93871,97561,102929, %U A298763 108127,112403,113341,114599,141937,144967,151883,151969,192883,224909,267961,270371,270577,270763,281531,282959,285979 %N A298763 Numbers that are the smallest of four consecutive primes, no three of which sum to a nonprime. %H A298763 Hans Havermann, Table of n, a(n) for n = 1..10000 %e A298763 19, 23, 29, 31 are four consecutive primes. The four ways of adding three of them yields 71, 73, 79, 83, all of which are prime. So 19 is a term of the sequence. %t A298763 s={2,3,5,7}; p=s[[-1]]; While[p<10^6, If[PrimeQ[s[[1]]+s[[2]]+s[[3]]]&&PrimeQ[s[[1]]+s[[2]]+s[[4]]]&&PrimeQ[s[[1]]+s[[3]]+s[[4]]]&&PrimeQ[s[[2]]+s[[3]]+s[[4]]], Print[s[[1]]]]; p=NextPrime[p]; s=Join[Rest[s],{p}]] %Y A298763 Subsequence of A073681. %K A298763 nonn,new %O A298763 1,1 %A A298763 _Hans Havermann_, Jan 26 2018 %I A298754 %S A298754 1,9,14,73,63,126,172,117,757,567,666,146,1099,1548,882,151,2457,6813, %T A298754 3430,219,2408,5994,6084,1638,15751,9891,1022,12556,12195,7938,14896, %U A298754 12483,9324,22113,10836,55261,25327,6174,15386,567,34461,21672,39754,16206,6813,54756,51912,2114,39331,47253 %N A298754 Numerator of sigma_3(n)/sigma_2(n). %p A298754 seq(numer(numtheory:-sigma[3](n)/numtheory:-sigma[2](n)),n=1..100); %o A298754 (PARI) a(n) = numerator(sigma(n, 3)/sigma(n, 2)); \\ _Michel Marcus_, Jan 27 2018 %Y A298754 Cf. A001158, A001157, A000203, A091258, A091259, A091260 (denominator). %K A298754 nonn,frac,new %O A298754 1,2 %A A298754 _Robert Israel_, Jan 26 2018 %I A297933 %S A297933 1,2,3,4,6,7,5,11,14,15,8,12,23,30,31,9,13,28,47,62,63,10,19,29,60,95, %T A297933 126,127,16,22,39,61,124,191,254,255,17,24,46,79,125,252,383,510,511, %U A297933 18,25,55,94,159,253,508,767,1022,1023,20,26,56,111,190,319 %N A297933 Rectangular array, by antidiagonals: row n gives the numbers whose base-2 digits d(m), d(m-1),...,d(0) having n as maximal run-length of 1s. %C A297933 Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers. %e A297933 Northwest corner: %e A297933 1 2 4 5 8 9 10 16 %e A297933 3 6 11 12 13 19 22 24 %e A297933 7 14 23 28 29 39 46 55 %e A297933 15 30 47 60 61 79 94 111 %e A297933 31 62 95 124 125 159 190 223 %e A297933 63 126 191 252 253 319 382 447 %e A297933 127 254 383 508 509 639 766 895 %e A297933 *** %e A297933 Base-2 digits of 59: 1,1,1,0,1,1 with runs 111 and 11 of 1s, so that 59 is in row 3. %t A297933 b = 2; s[n_] := Split[IntegerDigits[n, b]]; %t A297933 m[n_, d_] := Union[Select[s[n], MemberQ[#, d] &]] %t A297933 h[n_, d_] := Max[Map[Length, m[n, d]]] %t A297933 z = 6000; w = t[d_] := Table[h[n, d], {n, 1, z}] /. -Infinity -> 0 %t A297933 TableForm[Table[Flatten[Position[t[1], d]], {d, 0, 8}]] (* A297933 array *) %t A297933 u[d_] := Flatten[Position[t[1], d]] %t A297933 v[d_, n_] := u[d][[n]]; %t A297933 Table[v[n, k - n + 1], {k, 1, 11}, {n, 1, k}] // Flatten (* A297933 sequence *) %Y A297933 Cf. A297769, A297932. %K A297933 nonn,base,easy,tabl,new %O A297933 1,2 %A A297933 _Clark Kimberling_, Jan 26 2018 %I A297932 %S A297932 1,3,2,7,5,4,15,6,9,8,31,10,12,17,16,63,11,18,24,33,32,127,13,19,34, %T A297932 48,65,64,255,14,20,35,66,96,129,128,511,21,25,40,67,130,192,257,256, %U A297932 1023,22,28,49,80,131,258,384,513,512,2047,23,36,56,97,160,259 %N A297932 Rectangular array, by antidiagonals: row n gives the numbers whose base-2 digits d(m), d(m-1),...,d(0) having n as maximal run-length of 0s. %C A297932 Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers. %e A297932 Northwest corner: %e A297932 1 3 7 15 31 63 127 %e A297932 2 5 6 10 11 13 14 %e A297932 4 9 12 18 19 20 25 %e A297932 8 17 24 34 35 40 49 %e A297932 16 33 48 66 67 80 97 %e A297932 32 65 96 130 131 160 193 %e A297932 *** %e A297932 Base-2 digits of 72: 1,0,0,1,0,0,0 with runs 00 and 000 of 0s, so that 72 is in row 3. %t A297932 b = 2; s[n_] := Split[IntegerDigits[n, b]]; %t A297932 m[n_, d_] := Union[Select[s[n], MemberQ[#, d] &]] %t A297932 h[n_, d_] := Max[Map[Length, m[n, d]]] %t A297932 z = 6000; w = t[d_] := Table[h[n, d], {n, 1, z}] /. -Infinity -> 0 %t A297932 TableForm[Table[Flatten[Position[t[0], d]], {d, 0, 8}]] (* A297932 array *) %t A297932 u[d_] := Flatten[Position[t[0], d]] %t A297932 v[d_, n_] := u[d][[n]]; %t A297932 Table[v[n, k - n + 1], {k, 0, 11}, {n, 0, k}] // Flatten (* A297932 sequence *) %Y A297932 Cf. A297769, A297933. %K A297932 nonn,base,easy,tabl,new %O A297932 1,2 %A A297932 _Clark Kimberling_, Jan 26 2018 %I A298787 %S A298787 1,4,11,21,34,51,71,94,121,151,184,221,261,304,351,401,454,511,571, %T A298787 634,701,771,844,921,1001,1084,1171,1261,1354,1451,1551,1654,1761, %U A298787 1871,1984,2101,2221,2344,2471,2601,2734,2871,3011,3154,3301,3451,3604,3761,3921,4084,4251,4421,4594,4771 %N A298787 Partial sums of A298786. %H A298787 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298787 Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1). %F A298787 G.f.: (x^4 + 2*x^3 + 4*x^2 + 2*x + 1) / ((1 - x)^2*(1 - x^3)). %F A298787 a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4. - _Colin Barker_, Jan 27 2018 %o A298787 (PARI) Vec((1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)) + O(x^60)) \\ _Colin Barker_, Jan 27 2018 %Y A298787 Cf. A298786. %K A298787 nonn,easy,new %O A298787 0,2 %A A298787 _N. J. A. Sloane_, Jan 26 2018 %I A298785 %S A298785 1,5,11,21,35,51,71,95,121,151,185,221,261,305,351,401,455,511,571, %T A298785 635,701,771,845,921,1001,1085,1171,1261,1355,1451,1551,1655,1761, %U A298785 1871,1985,2101,2221,2345,2471,2601,2735,2871,3011,3155,3301,3451,3605,3761,3921,4085,4251,4421,4595,4771 %N A298785 Partial sums of A298784. %H A298785 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298785 Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1). %F A298785 G.f.: (1 + x^2)*(1 + 3*x + x^2) / ((1 - x)^2*(1 - x^3)). %F A298785 a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4. - _Colin Barker_, Jan 27 2018 %o A298785 (PARI) Vec((1 + x^2)*(1 + 3*x + x^2) / ((1 - x)^3*(1 + x + x^2)) + O(x^60)) \\ _Colin Barker_, Jan 27 2018 %Y A298785 Cf. A298784. %K A298785 nonn,easy,new %O A298785 0,2 %A A298785 _N. J. A. Sloane_, Jan 26 2018 %I A298786 %S A298786 1,3,7,10,13,17,20,23,27,30,33,37,40,43,47,50,53,57,60,63,67,70,73,77, %T A298786 80,83,87,90,93,97,100,103,107,110,113,117,120,123,127,130,133,137, %U A298786 140,143,147,150,153,157,160,163,167,170,173,177,180,183,187,190,193,197,200,203,207,210,213,217 %N A298786 Expansion of (x^4 + 2*x^3 + 4*x^2 + 2*x + 1) / ((1 - x)*(1 - x^3)). %C A298786 Appears to be the coordination sequence for a trivalent node in the bex tiling (or net). %H A298786 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298786 Reticular Chemistry Structure Resource (RCSR), The bex tiling (or net) %H A298786 Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1). %F A298786 a(n) = a(n-1) + a(n-3) - a(n-4) for n>4. - _Colin Barker_, Jan 27 2018 %p A298786 f3:=proc(n) %p A298786 if n=0 then 1 %p A298786 elif (n mod 3) = 0 then 10*n/3 %p A298786 elif (n mod 3) = 1 then (10*n-1)/3 %p A298786 else (10*n+1)/3; fi; end; %p A298786 [seq(f3(n),n=0..80)]; %o A298786 (PARI) Vec((1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x)^2*(1 + x + x^2)) + O(x^100)) \\ _Colin Barker_, Jan 27 2018 %Y A298786 Cf. A298784, A298787. %K A298786 nonn,easy,new %O A298786 0,2 %A A298786 _N. J. A. Sloane_, Jan 26 2018 %I A298784 %S A298784 1,4,6,10,14,16,20,24,26,30,34,36,40,44,46,50,54,56,60,64,66,70,74,76, %T A298784 80,84,86,90,94,96,100,104,106,110,114,116,120,124,126,130,134,136, %U A298784 140,144,146,150,154,156,160,164,166,170,174,176,180,184,186,190,194,196,200,204,206,210,214,216,220 %N A298784 Expansion of (1 + x^2)*(1 + 3*x + x^2) / ((1 - x)*(1 - x^3)). %C A298784 Appears to be the coordination sequence for a tetravalent node in the bex tiling (or net). %H A298784 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298784 Reticular Chemistry Structure Resource (RCSR), The bex tiling (or net) %H A298784 Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1). %F A298784 a(0)=1; thereafter, a(3*k) = 10*k, a(3*k+1) = 10*k+4, a(3*k+2) = 10*k+6. %F A298784 a(n) = a(n-1) + a(n-3) - a(n-4) for n>4. - _Colin Barker_, Jan 27 2018 %p A298784 f4:=proc(n) %p A298784 if n=0 then 1 %p A298784 elif (n mod 3) = 0 then 10*n/3 %p A298784 elif (n mod 3) = 1 then (10*n+2)/3 %p A298784 else (10*n-2)/3; fi; end; %p A298784 [seq(f4(n),n=0..80)]; %o A298784 (PARI) Vec((1 + x^2)*(1 + 3*x + x^2) / ((1 - x)^2*(1 + x + x^2)) + O(x^100)) \\ _Colin Barker_, Jan 27 2018 %Y A298784 Cf. A298785, A298786. %K A298784 nonn,easy,new %O A298784 0,2 %A A298784 _N. J. A. Sloane_, Jan 26 2018 %I A298686 %S A298686 13440,19440,19800,24480,49680,61560,104160,229320,298200,311040, %T A298686 329400,436800,471240,600600,1202040,1299600,1468800,1564920,1702800, %U A298686 2031120,2352240,2402400,2499840,2762760,2805600,2937600,2962080,3150840,3262680,3405600,3843840 %N A298686 Numbers i such that Fibonacci(i) is divisible by i+k for k=0,1,2,3,4. %C A298686 A subsequence of A298685. %H A298686 Chai Wah Wu, Table of n, a(n) for n = 1..147 %o A298686 (PARI) isone(n, k) = !(fibonacci(n) % (n+k)); %o A298686 isok(n) = isone(n,0) && isone(n,1) && isone(n,2) && isone(n,3) && isone(n,4); \\ _Michel Marcus_, Jan 28 2018 %Y A298686 Cf. A000045, A023172, A217738, A221018, A225219, A298684, A298685. %K A298686 nonn,new %O A298686 1,1 %A A298686 _Alex Ratushnyak_, Jan 24 2018 %E A298686 More terms from _Alois P. Heinz_, Jan 25 2018 %I A298688 %S A298688 1,60,60,540,13440,13440,329400,175472640 %N A298688 a(n) is the least i such that Fibonacci(i) is divisible by i+k for all k=0..n. %C A298688 a(0)..a(5) are first terms of sequences A023172, A217738, A298684, A298685, A298686, A298687. %e A298688 Least i such that Fibonacci(i) is divisible by i and i+1 (cf. A217738) is i=60, therefore a(1)=60. %e A298688 Least i such that Fibonacci(i) is divisible by i, i+1, ..., i+5 (cf. A298687) is i=13440, therefore a(5)=13440. %Y A298688 Cf. A000045, A023172, A217738, A221018, A225219, A298684, A298685, A298686, A298687. %K A298688 nonn,hard,more,new %O A298688 0,2 %A A298688 _Alex Ratushnyak_, Jan 24 2018 %E A298688 a(7) from _Alois P. Heinz_, Jan 25 2018 %E A298688 a(6) inserted by _Chai Wah Wu_, Jan 26 2018 %I A298700 %S A298700 1,6,25,120,581,2877,14421,72996,372229,1909336,9840909,50923041, %T A298700 264391973,1376654747,7185811685,37589283916,197005160825, %U A298700 1034244838815,5437798710585,28629290831670,150913830095445,796396974477495,4206974157985845,22243990866224505 %N A298700 a(n) = (n/2)*Sum_{k=1..n} C(n + k, n)*C(k, n - k)/k. %p A298700 a := n -> (n/2)*add(binomial(n + k, n)*binomial(k, n - k)/k, k=1..n): %p A298700 seq(a(n), n=1..24); %p A298700 # Alternatively: %p A298700 a := n -> `if`(n mod 2=0, 1, n/2)*binomial(2*n - floor(n/2), ceil(n/2))*hypergeom( %p A298700 [-floor(n/2), ceil(n/2), floor(3*(n+1)/2)], [n mod 2+1/2, ceil(n/2)+1], -1/4): %p A298700 seq(simplify(a(n)), n=1..24); %o A298700 (PARI) a(n) = (n/2)*sum(k=1, n, binomial(n+k, n)*binomial(k, n-k)/k); \\ _Michel Marcus_, Jan 27 2018 %K A298700 nonn,new %O A298700 1,2 %A A298700 _Peter Luschny_, Jan 26 2018 %I A298610 %S A298610 1,0,1,2,0,3,0,12,0,10,10,0,60,0,35,0,105,0,280,0,126,56,0,756,0,1260, %T A298610 0,462,0,840,0,4620,0,5544,0,1716,330,0,7920,0,25740,0,24024,0,6435,0, %U A298610 6435,0,60060,0,135135,0,102960,0,24310 %N A298610 Triangle read by rows, the unsigned coefficients of G(n, n, x/2) where G(n,a,x) denotes the n-th Gegenbauer polynomial, T(n, k) for 0 <= k <= n. %F A298610 G(n, x) = binomial(3*n-1, n)*hypergeom([-n, 3*n], [n+1/2], 1/2 - x/4). %e A298610 [0] 1 %e A298610 [1] 0, 1 %e A298610 [2] 2, 0, 3 %e A298610 [3] 0, 12, 0, 10 %e A298610 [4] 10, 0, 60, 0, 35 %e A298610 [5] 0, 105, 0, 280, 0, 126 %e A298610 [6] 56, 0, 756, 0, 1260, 0, 462 %e A298610 [7] 0, 840, 0, 4620, 0, 5544, 0, 1716 %e A298610 [8] 330, 0, 7920, 0, 25740, 0, 24024, 0, 6435 %e A298610 [9] 0, 6435, 0, 60060, 0, 135135, 0, 102960, 0, 24310 %p A298610 with(orthopoly): %p A298610 seq(seq((-1)^iquo(n-k, 2)*coeff(G(n,n,x/2),x,k), k=0..n), n=0..9); %t A298610 p[n_] := Binomial[3 n - 1, n] Hypergeometric2F1[-n, 3 n, n + 1/2, 1/2 - x/4]; %t A298610 Flatten[Table[(-1)^Floor[(n-k)/2] Coefficient[p[n], x, k], {n,0,9}, {k,0,n}]] %Y A298610 T(2n, 0) = A165817(n). T(n,n) = A088218(n). Row sums are A213684. %Y A298610 Cf. A109187. %K A298610 nonn,tabl,new %O A298610 0,4 %A A298610 _Peter Luschny_, Jan 25 2018 %I A298734 %S A298734 1,1,3,4,5,3,7,1,1,5,11,1,13,7,3,16,17,1,19,10,21,11,23,1,25,13,3,4, %T A298734 29,3,31,16,33,17,5,1,37,19,3,1,41,21,43,22,9,23,47,3,49,25,3,4,53,3, %U A298734 5,1,57,29,59,1,61,31,9,64,65,33,67,34,69,5,71,1,73,37,15,4,77,3,79,1,81,41,83,1,85,43,3,1,89,10,7,46,93,47 %N A298734 a(n) = n-th term in periodic sequence repeating the divisors of n in decreasing order. %H A298734 Alois P. Heinz, Table of n, a(n) for n = 1..20000 %e A298734 The divisors of 6 are 1, 2, 3, 6, which reversed is 6,3,2,1; repeating that produces the sequence 6, 3, 2, 1, 6, 3, 2, 1, 6, 3, 2, 1, ...; the 6th term in that sequence is 3, so a(6) = 3. %p A298734 with(numtheory): %p A298734 a:= n-> n/(l-> l[1+irem(n-1, nops(l))])(sort([divisors(n)[]])): %p A298734 seq(a(n), n=1..100); # _Alois P. Heinz_, Jan 29 2018 %o A298734 (PARI) a(n) = my(d=Vecrev(divisors(n))); if (n % #d, d[n % #d], 1); \\ _Michel Marcus_, Jan 26 2018 %Y A298734 Cf. A122377 (n/a(n)), A033950 (indices of 1's). %K A298734 nonn,new %O A298734 1,3 %A A298734 _Franklin T. Adams-Watters_, Jan 25 2018 %I A298783 %S A298783 0,0,0,0,1,0,1,1,1,1,2,1,2,2,2,2,3,2,3,3,3,3,4,3,4,4,4,4,5,7,5,5,5,6, %T A298783 6,5,6,7,6,6,7,7,7,7,7,8,10,7,8,9,8,8,9,9,10,9,9,10,10,9 %N A298783 Number of exceptional cases in an analysis of the muffin problem with n students. %H A298783 Guangiqi Cui et al., The Muffin Problem, arXiv:1709.02452 [math.CO], 2017. See Appendix G. %K A298783 nonn,more,new %O A298783 1,11 %A A298783 _N. J. A. Sloane_, Jan 26 2018 %I A298746 %S A298746 2,3,5,7,10,11,13,14,15,17,18,19,21,22,23,26,27,29,30,31,34,35,37,39, %T A298746 41,42,43,45,46,47,50,51,53,54,55,58,59,61,62,63,66,67,69,70,71,73,74, %U A298746 75,77,78,79,81,82,83,85,86,87,89,90,91,93,94,95,97,98,99,101,103,106,107,109,110,111,113,114,115,117,118 %N A298746 Numbers n whose base-2 representation can be written as the concatenation of the base-2 representations of a list of prime numbers, allowing leading zeros. %C A298746 Has positive density in the positive integers, which can be proved by considering only those numbers obtained using the primes 2 and 3. %e A298746 For example, 26 is such a number because 26 in base 2 is 11010, which can be written as the concatenation (11)(010). %Y A298746 Cf. A090421, which does not allow leading zeros. %K A298746 nonn,base,new %O A298746 1,1 %A A298746 _Jeffrey Shallit_, Jan 25 2018 %I A298782 %S A298782 1,1,1,1,1,1,1,1,1,1,1,2,6,2,1,1,5,2,2,5,1,1,9,4,4,4,9,1,1,22,7,11,11, %T A298782 7,22,1,1,45,8,26,37,26,8,45,1,1,101,15,76,178,178,76,15,101,1,1,218, %U A298782 23,222,139,562,139,222,23,218,1,1,477,38,721,823,2607,2607,823,721,38,477 %N A298782 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298782 Table starts %C A298782 .1...1..1...1....1.....1......1........1.........1..........1............1 %C A298782 .1...1..1...2....5.....9.....22.......45.......101........218..........477 %C A298782 .1...1..6...2....4.....7......8.......15........23.........38...........61 %C A298782 .1...2..2...4...11....26.....76......222.......721.......2361.........7737 %C A298782 .1...5..4..11...37...178....139......823......3906......12610........55215 %C A298782 .1...9..7..26..178...562...2607....15018.....81636.....455586......2645277 %C A298782 .1..22..8..76..139..2607...9889....75446....868689....7675611.....76328995 %C A298782 .1..45.15.222..823.15018..75446..1477645..18375492..261985685...3562741942 %C A298782 .1.101.23.721.3906.81636.868689.18375492.403292040.8471002968.176635665084 %H A298782 R. H. Hardin, Table of n, a(n) for n = 1..219 %F A298782 Empirical for column k: %F A298782 k=1: a(n) = a(n-1) %F A298782 k=2: a(n) = a(n-1) +3*a(n-2) -2*a(n-4) for n>5 %F A298782 k=3: a(n) = a(n-1) +a(n-2) for n>7 %F A298782 k=4: [order 21] for n>27 %e A298782 Some solutions for n=5 k=4 %e A298782 ..0..0..1..1. .0..0..1..1. .0..0..1..1. .0..0..1..1. .0..0..0..0 %e A298782 ..0..0..1..1. .0..0..1..1. .0..0..1..1. .0..0..1..1. .0..0..0..0 %e A298782 ..0..1..1..0. .0..0..1..1. .1..0..0..1. .1..0..1..1. .0..0..0..0 %e A298782 ..0..0..1..1. .1..1..0..0. .0..0..1..1. .0..0..1..1. .1..1..1..1 %e A298782 ..0..0..1..1. .1..1..0..0. .0..0..1..1. .0..0..1..1. .1..1..1..1 %Y A298782 Column 2 is A052962(n-2). %K A298782 nonn,tabl,new %O A298782 1,12 %A A298782 _R. H. Hardin_, Jan 26 2018 %I A298781 %S A298781 1,22,8,76,139,2607,9889,75446,868689,7675611,76328995,771602616, %T A298781 7685566410,77998227871,785605065928,7947378593295,80347627263007, %U A298781 812287066601605,8214879671287792,83066600919438416,840020904456433023 %N A298781 Number of nX7 0..1 arrays with every element equal to 0, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298781 Column 7 of A298782. %H A298781 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298781 Some solutions for n=5 %e A298781 ..0..0..1..0..0..0..0. .0..0..1..1..1..0..0. .0..0..1..1..1..0..0 %e A298781 ..0..0..0..0..0..0..0. .0..0..1..1..1..0..0. .0..0..1..1..1..0..0 %e A298781 ..1..1..1..1..1..1..1. .1..0..1..1..1..0..1. .1..1..0..0..0..1..1 %e A298781 ..1..1..1..1..1..1..1. .0..0..1..1..1..0..0. .1..1..0..0..0..1..1 %e A298781 ..1..1..1..1..1..1..1. .0..0..1..1..1..0..0. .1..1..0..0..0..1..1 %Y A298781 Cf. A298782. %K A298781 nonn,new %O A298781 1,2 %A A298781 _R. H. Hardin_, Jan 26 2018 %I A298780 %S A298780 1,9,7,26,178,562,2607,15018,81636,455586,2645277,15071152,86168841, %T A298780 496188667,2843376149,16332351166,93816373371,538686228311, %U A298780 3093975180196,17770633217448,102061638116874,586199211091864,3366879038649282 %N A298780 Number of nX6 0..1 arrays with every element equal to 0, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298780 Column 6 of A298782. %H A298780 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298780 Some solutions for n=5 %e A298780 ..0..0..0..0..0..1. .0..1..1..0..1..0. .0..1..1..1..1..0. .0..1..1..1..1..0 %e A298780 ..0..0..1..0..0..0. .1..1..1..1..1..1. .1..1..0..0..1..1. .1..1..0..0..1..1 %e A298780 ..0..1..1..1..0..1. .0..1..0..0..1..0. .0..1..0..0..1..0. .1..0..0..0..1..0 %e A298780 ..0..0..1..1..0..0. .1..1..0..0..1..1. .1..1..1..1..1..1. .1..1..0..1..1..1 %e A298780 ..0..0..0..0..0..0. .1..1..1..1..1..1. .0..1..1..0..1..1. .1..1..1..1..1..0 %Y A298780 Cf. A298782. %K A298780 nonn,new %O A298780 1,2 %A A298780 _R. H. Hardin_, Jan 26 2018 %I A298779 %S A298779 1,5,4,11,37,178,139,823,3906,12610,55215,258220,1095250,4916214, %T A298779 22212973,98542757,442037397,1983868610,8869475923,39754021798, %U A298779 178147007248,797763231571,3574347245858,16013491545305,71732996079806 %N A298779 Number of nX5 0..1 arrays with every element equal to 0, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298779 Column 5 of A298782. %H A298779 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298779 Some solutions for n=5 %e A298779 ..0..0..0..0..1. .0..0..0..0..0. .0..1..1..1..1. .0..0..1..0..0 %e A298779 ..0..0..1..0..0. .0..0..0..0..0. .1..1..0..1..1. .0..0..0..0..0 %e A298779 ..0..1..1..1..0. .0..0..0..0..0. .1..0..0..0..1. .1..1..1..1..1 %e A298779 ..0..0..1..0..0. .1..1..1..1..1. .1..1..0..1..1. .1..1..1..1..1 %e A298779 ..1..0..0..0..1. .1..1..0..1..1. .0..1..1..1..1. .1..1..1..1..1 %Y A298779 Cf. A298782. %K A298779 nonn,new %O A298779 1,2 %A A298779 _R. H. Hardin_, Jan 26 2018 %I A298778 %S A298778 1,2,2,4,11,26,76,222,721,2361,7737,25780,85449,284610,947298,3154457, %T A298778 10507280,34994885,116569493,388281532,1293363746,4308190293, %U A298778 14350558511,47801810756,159227933912,530389181794,1766729082040,5884984209038 %N A298778 Number of nX4 0..1 arrays with every element equal to 0, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298778 Column 4 of A298782. %H A298778 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298778 Empirical: a(n) = 3*a(n-1) +5*a(n-2) -8*a(n-3) -21*a(n-4) +8*a(n-5) +16*a(n-6) +19*a(n-7) +5*a(n-8) +10*a(n-9) -12*a(n-10) -43*a(n-11) -12*a(n-12) -3*a(n-13) +26*a(n-14) +11*a(n-15) +7*a(n-16) -3*a(n-17) -13*a(n-18) +7*a(n-19) +2*a(n-20) -3*a(n-21) for n>27 %e A298778 Some solutions for n=5 %e A298778 ..0..0..1..1. .0..0..1..1. .0..0..0..0. .0..0..1..1. .0..0..0..0 %e A298778 ..0..0..1..1. .0..0..1..1. .0..0..0..0. .0..0..1..1. .0..0..0..0 %e A298778 ..1..1..0..0. .0..0..1..0. .0..0..0..0. .1..0..0..1. .0..0..0..0 %e A298778 ..1..1..0..0. .0..0..1..1. .0..0..0..0. .0..0..1..1. .1..1..1..1 %e A298778 ..1..1..0..0. .0..0..1..1. .0..0..0..0. .0..0..1..1. .1..1..1..1 %Y A298778 Cf. A298782. %K A298778 nonn,new %O A298778 1,2 %A A298778 _R. H. Hardin_, Jan 26 2018 %I A298777 %S A298777 1,1,6,2,4,7,8,15,23,38,61,99,160,259,419,678,1097,1775,2872,4647, %T A298777 7519,12166,19685,31851,51536,83387,134923,218310,353233,571543, %U A298777 924776,1496319,2421095,3917414,6338509,10255923,16594432,26850355,43444787 %N A298777 Number of nX3 0..1 arrays with every element equal to 0, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298777 Column 3 of A298782. %H A298777 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298777 Empirical: a(n) = a(n-1) +a(n-2) for n>7 %e A298777 All solutions for n=5 %e A298777 ..0..1..0. .0..0..0. .0..0..0. .0..0..0 %e A298777 ..1..1..1. .0..0..0. .0..0..0. .0..0..0 %e A298777 ..1..0..1. .0..0..0. .0..0..0. .1..1..1 %e A298777 ..1..1..1. .0..0..0. .1..1..1. .1..1..1 %e A298777 ..0..1..0. .0..0..0. .1..1..1. .1..1..1 %Y A298777 Cf. A298782. %K A298777 nonn,new %O A298777 1,3 %A A298777 _R. H. Hardin_, Jan 26 2018 %I A298776 %S A298776 1,1,6,4,37,562,9889,1477645,403292040,309941364263 %N A298776 Number of nXn 0..1 arrays with every element equal to 0, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298776 Diagonal of A298782. %e A298776 Some solutions for n=5 %e A298776 ..0..0..0..1..1. .0..1..1..1..1. .0..0..1..0..0. .0..0..1..0..1 %e A298776 ..0..0..0..1..1. .1..1..0..1..1. .0..0..0..0..0. .0..0..0..0..0 %e A298776 ..1..1..1..0..0. .1..0..0..0..1. .1..1..1..1..1. .1..1..1..0..1 %e A298776 ..1..1..1..0..0. .1..1..0..1..1. .1..1..1..1..1. .1..1..1..0..0 %e A298776 ..1..1..1..0..0. .1..1..1..1..1. .1..1..1..1..1. .1..1..1..0..0 %Y A298776 Cf. A298782. %K A298776 nonn,new %O A298776 1,3 %A A298776 _R. H. Hardin_, Jan 26 2018 %I A298747 %S A298747 1,2,3,4,5,6,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26, %T A298747 27,28,29,30,31,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81, %U A298747 82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108 %N A298747 If n = 2^(2*m+1)+j, 0 <= j < 3*2^(2*m+1), then a(n) = 2^(3*m)+j. %C A298747 An easily computed sequence, growing faster than linearly, that provably contains infinitely many primes. %H A298747 R. Israel, Answer to "What's the fastest growing function known to contain infinitely many primes?", Mathematics StackExchange. %F A298747 G.f.: x^2/(1-x)^2 + Sum_{m>=1} (7*2^(3*m-3)-3*2^(2*m-1))*x^(2*4^m)/(1-x). %p A298747 f:= proc(n) local m,j; %p A298747 m:= floor(log[4](n/2)); %p A298747 2^(3*m)+n - 2^(2*m+1) %p A298747 end proc: %p A298747 map(f, [$2..100]); # _Robert Israel_, Jan 25 2018 %K A298747 nonn,easy,new %O A298747 2,2 %A A298747 _Robert Israel_, Jan 25 2018 %I A298775 %S A298775 0,1,1,1,3,1,2,7,7,2,3,13,15,13,3,5,23,25,25,23,5,8,49,47,78,47,49,8, %T A298775 13,99,109,233,233,109,99,13,21,189,245,779,682,779,245,189,21,34,383, %U A298775 545,2359,2603,2603,2359,545,383,34,55,777,1253,7024,11657,11320,11657,7024 %N A298775 T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298775 Table starts %C A298775 ..0...1....1.....2......3.......5........8........13.........21..........34 %C A298775 ..1...3....7....13.....23......49.......99.......189........383.........777 %C A298775 ..1...7...15....25.....47.....109......245.......545.......1253........2859 %C A298775 ..2..13...25....78....233.....779.....2359......7024......21572.......66763 %C A298775 ..3..23...47...233....682....2603....11657.....39908.....149791......617528 %C A298775 ..5..49..109...779...2603...11320....61333....284618....1376511.....6959439 %C A298775 ..8..99..245..2359..11657...61333...484813...2887678...18497464...127977516 %C A298775 .13.189..545..7024..39908..284618..2887678..22160831..186728055..1683257109 %C A298775 .21.383.1253.21572.149791.1376511.18497464.186728055.2093453587.24925712914 %H A298775 R. H. Hardin, Table of n, a(n) for n = 1..219 %F A298775 Empirical for column k: %F A298775 k=1: a(n) = a(n-1) +a(n-2) %F A298775 k=2: a(n) = 2*a(n-1) -a(n-2) +4*a(n-3) -4*a(n-4) for n>5 %F A298775 k=3: [order 12] for n>13 %F A298775 k=4: [order 62] for n>65 %e A298775 Some solutions for n=6 k=5 %e A298775 ..0..0..0..1..0. .0..1..0..0..0. .0..1..1..1..0. .0..0..0..0..0 %e A298775 ..1..0..0..0..1. .1..0..0..0..1. .1..0..0..0..1. .1..0..0..0..1 %e A298775 ..0..1..1..1..0. .0..1..1..1..0. .1..0..0..0..1. .0..1..1..1..0 %e A298775 ..0..1..1..1..0. .0..1..1..1..0. .1..0..0..0..1. .0..1..1..1..0 %e A298775 ..0..1..1..1..0. .0..1..1..1..0. .0..1..1..1..0. .0..1..1..1..0 %e A298775 ..1..0..0..0..1. .1..0..0..0..1. .1..1..1..0..1. .1..0..0..0..1 %Y A298775 Column 1 is A000045(n-1). %Y A298775 Column 2 is A297953. %Y A298775 Column 3 is A297954. %Y A298775 Column 4 is A297955. %K A298775 nonn,tabl,new %O A298775 1,5 %A A298775 _R. H. Hardin_, Jan 26 2018 %I A298774 %S A298774 8,99,245,2359,11657,61333,484813,2887678,18497464,127977516, %T A298774 830209200,5490680497,36705295118,242989778508,1615075659957, %U A298774 10732935729937,71283281214068,473870950797066,3148048592750846,20916953345403645 %N A298774 Number of nX7 0..1 arrays with every element equal to 1, 2, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298774 Column 7 of A298775. %H A298774 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298774 Some solutions for n=6 %e A298774 ..0..1..0..0..0..0..1. .0..1..1..1..1..1..0. .0..1..0..0..0..0..1 %e A298774 ..0..0..1..1..1..1..0. .1..0..0..0..0..0..1. .1..0..0..0..1..0..1 %e A298774 ..0..0..1..1..1..1..0. .1..0..0..0..0..0..1. .0..1..1..1..0..0..1 %e A298774 ..0..0..1..1..1..1..0. .1..0..0..0..0..0..1. .0..1..1..1..0..0..1 %e A298774 ..0..0..1..1..1..1..0. .0..1..1..1..1..1..0. .0..1..1..1..0..0..1 %e A298774 ..0..1..0..0..0..0..1. .1..0..1..1..1..1..1. .1..0..0..0..1..1..0 %Y A298774 Cf. A298775. %K A298774 nonn,new %O A298774 1,1 %A A298774 _R. H. Hardin_, Jan 26 2018 %I A298773 %S A298773 5,49,109,779,2603,11320,61333,284618,1376511,6959439,34511708, %T A298773 170670413,851900683,4250558598,21161145459,105481131616,526133966100, %U A298773 2623203957029,13078950820473,65224465637861,325269426017123 %N A298773 Number of nX6 0..1 arrays with every element equal to 1, 2, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298773 Column 6 of A298775. %H A298773 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298773 Some solutions for n=7 %e A298773 ..0..1..0..0..1..0. .0..1..0..0..0..0. .0..1..1..1..0..1. .0..1..0..1..0..0 %e A298773 ..1..0..0..0..0..1. .1..0..0..0..0..1. .1..0..0..0..1..1. .1..0..0..1..0..1 %e A298773 ..0..1..1..1..1..0. .0..1..1..1..1..0. .1..0..0..0..1..1. .1..0..0..0..1..0 %e A298773 ..0..1..1..1..1..0. .0..1..1..1..1..0. .1..0..0..0..1..0. .0..1..1..1..0..1 %e A298773 ..0..1..1..1..1..0. .0..1..1..1..1..0. .0..1..1..1..0..1. .0..1..1..1..0..0 %e A298773 ..0..1..1..1..1..0. .1..0..0..0..0..1. .1..1..1..0..1..0. .0..1..1..1..0..0 %e A298773 ..1..0..0..0..0..1. .0..0..0..0..0..0. .0..0..1..0..1..1. .1..0..0..0..1..0 %Y A298773 Cf. A298775. %K A298773 nonn,new %O A298773 1,1 %A A298773 _R. H. Hardin_, Jan 26 2018 %I A298772 %S A298772 3,23,47,233,682,2603,11657,39908,149791,617528,2340328,8921969, %T A298772 34929827,135499003,522693251,2023538485,7856615385,30426642588, %U A298772 117738552981,456437251086,1768840207859,6850066781045,26539738521878,102838505746523 %N A298772 Number of nX5 0..1 arrays with every element equal to 1, 2, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298772 Column 5 of A298775. %H A298772 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298772 Some solutions for n=7 %e A298772 ..0..0..0..0..0. .0..1..1..1..0. .0..1..1..1..0. .0..1..1..1..0 %e A298772 ..1..0..0..0..1. .1..0..0..0..1. .1..0..0..0..1. .1..0..0..0..1 %e A298772 ..0..1..1..1..0. .1..0..0..0..1. .1..0..0..0..1. .1..0..0..0..1 %e A298772 ..0..1..1..1..0. .1..0..0..0..1. .1..0..0..0..1. .0..1..1..1..0 %e A298772 ..0..1..1..1..0. .0..1..1..1..0. .0..1..1..1..0. .0..1..1..1..0 %e A298772 ..1..0..0..0..1. .1..1..1..1..0. .1..1..1..0..1. .0..1..1..1..0 %e A298772 ..0..1..0..0..0. .0..0..0..0..1. .0..0..1..0..1. .1..0..0..0..1 %Y A298772 Cf. A298775. %K A298772 nonn,new %O A298772 1,1 %A A298772 _R. H. Hardin_, Jan 26 2018 %I A298771 %S A298771 0,3,15,78,682,11320,484813,22160831,2093453587,397340932476 %N A298771 Number of nXn 0..1 arrays with every element equal to 1, 2, 4, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298771 Diagonal of A298775. %e A298771 Some solutions for n=6 %e A298771 ..0..1..0..0..0..0. .0..1..0..0..0..1. .0..1..1..1..0..1. .0..0..0..0..0..0 %e A298771 ..1..0..0..0..1..0. .1..0..1..1..1..0. .1..0..0..0..1..1. .0..1..0..0..0..1 %e A298771 ..0..1..1..1..0..0. .0..0..1..1..1..0. .1..0..0..0..1..1. .0..0..1..1..1..0 %e A298771 ..0..1..1..1..0..0. .0..0..1..1..1..0. .1..0..0..0..1..0. .0..0..1..1..1..0 %e A298771 ..0..1..1..1..0..0. .0..1..0..0..0..1. .0..1..1..1..0..1. .1..0..1..1..1..0 %e A298771 ..1..0..0..0..1..0. .0..0..0..0..1..0. .1..1..1..0..1..0. .0..1..0..0..0..1 %Y A298771 Cf. A298775. %K A298771 nonn,new %O A298771 1,2 %A A298771 _R. H. Hardin_, Jan 26 2018 %I A298770 %S A298770 0,1,1,1,4,1,2,18,18,2,3,52,57,52,3,5,174,222,222,174,5,8,604,808,957, %T A298770 808,604,8,13,2048,3124,4288,4288,3124,2048,13,21,6948,11807,19722, %U A298770 23932,19722,11807,6948,21,34,23652,44846,89295,135243,135243,89295,44846 %N A298770 T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 3, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298770 Table starts %C A298770 ..0.....1......1.......2........3.........5..........8..........13...........21 %C A298770 ..1.....4.....18......52......174.......604.......2048........6948........23652 %C A298770 ..1....18.....57.....222......808......3124......11807.......44846.......170350 %C A298770 ..2....52....222.....957.....4288.....19722......89295......406426......1851746 %C A298770 ..3...174....808....4288....23932....135243.....754713.....4245549.....23848195 %C A298770 ..5...604...3124...19722...135243....942727....6456802....44530673....307427453 %C A298770 ..8..2048..11807...89295...754713...6456802...53950338...454801194...3847185453 %C A298770 .13..6948..44846..406426..4245549..44530673..454801194..4698205702..48730992802 %C A298770 .21.23652.170350.1851746.23848195.307427453.3847185453.48730992802.620363350692 %H A298770 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A298770 Empirical for column k: %F A298770 k=1: a(n) = a(n-1) +a(n-2) %F A298770 k=2: a(n) = 4*a(n-1) -2*a(n-2) +2*a(n-3) -6*a(n-4) -4*a(n-5) for n>6 %F A298770 k=3: [order 19] for n>20 %F A298770 k=4: [order 66] for n>69 %e A298770 Some solutions for n=5 k=4 %e A298770 ..0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..0 %e A298770 ..1..1..1..0. .1..1..1..1. .0..0..0..0. .1..1..1..1. .0..0..0..0 %e A298770 ..1..1..1..0. .0..0..0..0. .0..0..0..0. .1..1..1..1. .1..1..1..1 %e A298770 ..1..1..1..0. .0..0..0..0. .1..1..1..1. .1..1..1..1. .1..1..1..1 %e A298770 ..1..1..1..0. .0..0..0..0. .0..0..0..1. .1..1..1..1. .1..1..1..1 %Y A298770 Column 1 is A000045(n-1). %Y A298770 Column 2 is A297945. %K A298770 nonn,tabl,new %O A298770 1,5 %A A298770 _R. H. Hardin_, Jan 26 2018 %I A298769 %S A298769 8,2048,11807,89295,754713,6456802,53950338,454801194,3847185453, %T A298769 32497437189,274558538901,2321648021996,19629628158181, %U A298769 165957305627364,1403226073953134,11865080326336066,100325233293376853 %N A298769 Number of nX7 0..1 arrays with every element equal to 1, 2, 3, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298769 Column 7 of A298770. %H A298769 R. H. Hardin, Table of n, a(n) for n = 1..112 %e A298769 Some solutions for n=4 %e A298769 ..0..0..0..1..1..0..0. .0..0..0..0..1..0..1. .0..0..1..0..0..0..1 %e A298769 ..0..0..0..1..0..1..0. .1..1..1..0..1..1..0. .0..0..1..0..0..0..1 %e A298769 ..0..0..0..1..0..1..1. .1..1..1..0..1..0..1. .0..0..1..0..0..0..1 %e A298769 ..0..0..0..1..0..0..1. .1..1..1..0..0..1..0. .0..0..1..0..0..0..1 %Y A298769 Cf. A298770. %K A298769 nonn,new %O A298769 1,1 %A A298769 _R. H. Hardin_, Jan 26 2018 %I A298768 %S A298768 5,604,3124,19722,135243,942727,6456802,44530673,307427453,2120354111, %T A298768 14630695675,101015435108,697302434516,4813188671290,33227405781627, %U A298768 229381447522706,1583479906813958,10931377863845814,75463796580783735 %N A298768 Number of nX6 0..1 arrays with every element equal to 1, 2, 3, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298768 Column 6 of A298770. %H A298768 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298768 Some solutions for n=5 %e A298768 ..0..1..1..1..0..1. .0..1..1..1..1..0. .0..0..0..0..0..0. .0..1..1..1..0..1 %e A298768 ..1..0..0..1..0..1. .1..0..0..1..0..1. .0..0..0..0..0..0. .0..1..1..1..0..1 %e A298768 ..1..0..1..1..1..0. .1..0..1..1..0..1. .0..0..0..0..0..0. .0..1..1..1..0..1 %e A298768 ..0..0..0..1..1..0. .0..0..0..0..1..0. .1..1..1..1..1..1. .0..1..1..1..0..1 %e A298768 ..1..1..1..0..0..0. .1..1..1..1..0..0. .0..0..0..0..0..0. .0..1..1..1..0..1 %Y A298768 Cf. A298770. %K A298768 nonn,new %O A298768 1,1 %A A298768 _R. H. Hardin_, Jan 26 2018 %I A298767 %S A298767 3,174,808,4288,23932,135243,754713,4245549,23848195,133798314, %T A298767 751604067,4222577179,23714049081,133198003207,748199961140, %U A298767 4202513392216,23605201896388,132590921168209,744758138601915,4183280457347730 %N A298767 Number of nX5 0..1 arrays with every element equal to 1, 2, 3, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298767 Column 5 of A298770. %H A298767 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298767 Some solutions for n=5 %e A298767 ..0..0..0..0..0. .0..0..0..0..1. .0..0..1..1..1. .0..0..1..1..1 %e A298767 ..1..1..1..1..1. .0..0..0..0..1. .1..0..1..1..1. .1..0..1..1..1 %e A298767 ..1..0..0..0..0. .0..0..0..0..1. .1..0..1..1..1. .1..0..1..1..1 %e A298767 ..1..0..0..0..0. .0..0..0..0..1. .1..0..0..0..0. .1..0..0..0..0 %e A298767 ..1..0..0..0..0. .1..1..1..1..1. .0..1..1..1..0. .1..1..1..1..1 %Y A298767 Cf. A298770. %K A298767 nonn,new %O A298767 1,1 %A A298767 _R. H. Hardin_, Jan 26 2018 %I A298766 %S A298766 2,52,222,957,4288,19722,89295,406426,1851746,8431005,38382433, %T A298766 174793642,795928404,3624236849,16503296767,75149172909,342196469101, %U A298766 1558217168200,7095460966718,32309708234520,147124667839968,669943229894924 %N A298766 Number of nX4 0..1 arrays with every element equal to 1, 2, 3, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298766 Column 4 of A298770. %H A298766 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298766 Empirical: a(n) = 10*a(n-1) -38*a(n-2) +105*a(n-3) -336*a(n-4) +811*a(n-5) -1377*a(n-6) +2650*a(n-7) -3981*a(n-8) +2010*a(n-9) +1813*a(n-10) -14149*a(n-11) +47479*a(n-12) -87794*a(n-13) +154393*a(n-14) -249813*a(n-15) +293618*a(n-16) -325584*a(n-17) +254489*a(n-18) +196658*a(n-19) -968191*a(n-20) +2375306*a(n-21) -4862588*a(n-22) +7851537*a(n-23) -11495750*a(n-24) +15631765*a(n-25) -17962600*a(n-26) +18511235*a(n-27) -16755230*a(n-28) +10681320*a(n-29) -4284093*a(n-30) -3288711*a(n-31) +12217870*a(n-32) -13556672*a(n-33) +14598558*a(n-34) -17004407*a(n-35) +7774093*a(n-36) -4009403*a(n-37) +9866848*a(n-38) -3203438*a(n-39) +3961299*a(n-40) -14654649*a(n-41) +7334662*a(n-42) -1791551*a(n-43) +3980858*a(n-44) +3597567*a(n-45) -6545193*a(n-46) +2366784*a(n-47) -2366081*a(n-48) +1799656*a(n-49) +326625*a(n-50) -402654*a(n-51) -202168*a(n-52) +198284*a(n-53) -67224*a(n-54) +127183*a(n-55) +1274*a(n-56) -38064*a(n-57) -10556*a(n-58) +1228*a(n-59) -4192*a(n-60) -994*a(n-61) -56*a(n-62) -422*a(n-63) -60*a(n-64) +32*a(n-66) for n>69 %e A298766 Some solutions for n=5 %e A298766 ..0..0..0..1. .0..1..1..1. .0..0..0..0. .0..1..1..0. .0..0..0..0 %e A298766 ..0..0..0..1. .0..1..1..1. .0..0..0..0. .0..0..0..0. .0..1..1..1 %e A298766 ..0..0..0..1. .0..1..1..1. .1..1..1..1. .1..1..1..1. .0..1..1..1 %e A298766 ..1..1..1..1. .0..1..1..1. .1..1..1..1. .1..1..1..1. .0..1..1..1 %e A298766 ..0..0..0..0. .0..1..1..1. .1..1..1..1. .1..1..1..1. .0..1..1..1 %Y A298766 Cf. A298770. %K A298766 nonn,new %O A298766 1,1 %A A298766 _R. H. Hardin_, Jan 26 2018 %I A298765 %S A298765 1,18,57,222,808,3124,11807,44846,170350,647730,2461372,9354803, %T A298765 35554194,135132441,513598421,1952048536,7419209212,28198435083, %U A298765 107174707010,407342469367,1548200080894,5884295781265,22364639638916,85002034082610 %N A298765 Number of nX3 0..1 arrays with every element equal to 1, 2, 3, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298765 Column 3 of A298770. %H A298765 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298765 Empirical: a(n) = 5*a(n-1) -4*a(n-2) -a(n-3) -3*a(n-4) -12*a(n-5) +33*a(n-6) -17*a(n-7) +3*a(n-8) -81*a(n-9) -4*a(n-10) +86*a(n-11) +127*a(n-12) +13*a(n-13) -95*a(n-14) -56*a(n-15) +8*a(n-16) +8*a(n-17) -16*a(n-18) -8*a(n-19) for n>20 %e A298765 Some solutions for n=5 %e A298765 ..0..0..0. .0..1..1. .0..1..0. .0..0..1. .0..1..1. .0..0..0. .0..1..0 %e A298765 ..1..1..0. .1..0..0. .0..1..0. .0..1..1. .0..1..0. .0..0..0. .0..1..0 %e A298765 ..1..0..1. .1..0..1. .0..1..1. .0..1..0. .0..1..0. .0..0..0. .1..1..0 %e A298765 ..0..1..0. .1..1..1. .1..0..0. .0..1..0. .1..0..0. .1..1..1. .0..1..0 %e A298765 ..0..1..0. .0..0..1. .1..1..1. .1..1..0. .0..1..0. .1..1..1. .0..0..1 %Y A298765 Cf. A298770. %K A298765 nonn,new %O A298765 1,2 %A A298765 _R. H. Hardin_, Jan 26 2018 %I A298764 %S A298764 0,4,57,957,23932,942727,53950338,4698205702,620363350692 %N A298764 Number of nXn 0..1 arrays with every element equal to 1, 2, 3, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298764 Diagonal of A298770. %e A298764 Some solutions for n=5 %e A298764 ..0..0..0..0..0. .0..1..1..1..0. .0..0..0..0..0. .0..1..0..0..0 %e A298764 ..1..1..1..1..1. .0..1..1..1..0. .0..1..1..1..0. .0..1..0..0..0 %e A298764 ..1..1..1..1..1. .0..1..1..1..0. .0..1..1..1..0. .0..1..0..0..0 %e A298764 ..1..1..1..1..1. .0..0..0..0..0. .0..1..1..1..0. .0..1..1..1..1 %e A298764 ..1..1..1..1..1. .1..1..1..1..1. .0..0..0..0..0. .0..0..0..0..1 %Y A298764 Cf. A298770. %K A298764 nonn,new %O A298764 1,2 %A A298764 _R. H. Hardin_, Jan 26 2018 %I A297351 %S A297351 1,2,3,4,6,6,7,8,9,10,10,11,12 %N A297351 Smallest number k such that, for any set S of k distinct nonzero residues mod p = prime(n), any residue mod p can be represented as a sum of zero or more distinct elements of S. %H A297351 P. Erdős and H. Heilbronn, On the addition of residue classes mod p, Acta Arithmetica 9 (1964), 149-159. %H A297351 John E. Olson, An addition theorem modulo p, Journal of Combinatorial Theory 5 (1968), pp. 45-52. %F A297351 For p = prime(n) > 3, sqrt(4p + 5) - 2 < a(n) <= sqrt(4p). The former bound is due to Erdős & Heilbronn and the latter to Olson. %o A297351 (PARI) sumHitsAll(v,m)=my(u=[0],n); for(i=1,#v, n=v[i]; u=Set(concat(u,apply(j->(j+n)%m,u))); if(#u==m, return(1))); 0 %o A297351 a(n,p=prime(n))=for(s=sqrtint(4*p+2)-1,sqrtint(4*p)-1, forvec(v=vector(s,i,[1,p-1]), if(!sumHitsAll(v,p), next(2)), 2); return(s)); sqrtint(4*p) %K A297351 hard,more,nonn,new %O A297351 1,2 %A A297351 _Charles R Greathouse IV_, Jan 24 2018 %E A297351 a(13) from _Charles R Greathouse IV_, Jan 27 2018 %I A298735 %S A298735 1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1, %T A298735 1,2,1,1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1, %U A298735 1,1,1,2,1,1,2,1,1,1,1,1,3,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,2,2,1,1,1,1,1 %N A298735 Number of odd squares dividing n. %F A298735 G.f.: Sum_{k>=1} x^((2*k-1)^2)/(1 - x^((2*k-1)^2)). %e A298735 a(81) = 3 because 81 has 5 divisors {1, 3, 9, 27, 81} among which 3 are odd squares {1, 9, 81}. %t A298735 nmax = 105; Rest[CoefficientList[Series[Sum[x^(2 k - 1)^2/(1 - x^(2 k - 1)^2), {k, 1, nmax}], {x, 0, nmax}], x]] %t A298735 a[n_] := Length[Select[Divisors[n], IntegerQ[Sqrt[#]] && OddQ[#] &]]; Table[a[n], {n, 1, 105}] %o A298735 (PARI) a(n)=factorback(apply(e->e\2+1, factor(n/2^valuation(n,2))[, 2])) \\ _Rémy Sigrist_, Jan 26 2018 %Y A298735 Cf. A001227, A016754, A046951, A056170, A071325, A122132 (positions of ones). %K A298735 nonn,mult,new %O A298735 1,9 %A A298735 _Ilya Gutkovskiy_, Jan 25 2018 %E A298735 Keyword mult added by _Rémy Sigrist_, Jan 26 2018 %I A298732 %S A298732 1,0,1,1,1,3,3,6,7,14,18,30,45,66,107,157,245,369,569,862,1325,2020, %T A298732 3078,4717,7183,10991,16769,25626,39117,59763,91264,139362,212893, %U A298732 325060,496525,758258,1158079,1768634,2701162,4125320,6300303,9622247,14695253,22443451,34276405,52348435 %N A298732 Number of compositions (ordered partitions) of n into parts > 1 such that no two adjacent parts are equal (Carlitz compositions). %H A298732 Alois P. Heinz, Table of n, a(n) for n = 0..1000 %H A298732 Index entries for sequences related to compositions %F A298732 G.f.: 1/(1 - Sum_{k>=2} x^k/(1 + x^k)). %e A298732 a(7) = 6 because we have [7], [5, 2], [4, 3], [3, 4], [2, 5] and [2, 3, 2]. %p A298732 b:= proc(n, i) option remember; `if`(n=0, 1, %p A298732 add(`if`(j=i, 0, b(n-j, `if`(j<=n-j, j, 0))), j=2..n)) %p A298732 end: %p A298732 a:= n-> b(n, 0): %p A298732 seq(a(n), n=0..50); # _Alois P. Heinz_, Jan 25 2018 %t A298732 nmax = 45; CoefficientList[Series[1/(1 - Sum[x^k/(1 + x^k), {k, 2, nmax}]), {x, 0, nmax}], x] %Y A298732 Cf. A000045, A002865, A003242, A011782, A025147, A032020, A114900, A155822, A212804, A218694. %K A298732 nonn,new %O A298732 0,6 %A A298732 _Ilya Gutkovskiy_, Jan 25 2018 %I A298672 %S A298672 1,1,0,0,0,0,20,0,1121,72828,872640,9037710,118590450,1743739426, %T A298672 24407782672,424735169040,7802802463460,135385454550288, %U A298672 2823521345232834,59332856029292241,1238888844244575904,28893281420537822022,684650546073054870188,16342742577592266281996 %N A298672 Number of ordered ways of writing n^3 as a sum of n positive cubes. %H A298672 Index entries for sequences related to sums of cubes %F A298672 a(n) = [x^(n^3)] (Sum_{k>=1} x^(k^3))^n. %e A298672 a(6) = 20 because we have [64, 64, 64, 8, 8, 8], [64, 64, 8, 64, 8, 8], [64, 64, 8, 8, 64, 8], [64, 64, 8, 8, 8, 64], [64, 8, 64, 64, 8, 8], [64, 8, 64, 8, 64, 8], [64, 8, 64, 8, 8, 64], [64, 8, 8, 64, 64, 8], [64, 8, 8, 64, 8, 64], [64, 8, 8, 8, 64, 64], [8, 64, 64, 64, 8, 8], [8, 64, 64, 8, 64, 8], [8, 64, 64, 8, 8, 64], [8, 64, 8, 64, 64, 8], [8, 64, 8, 64, 8, 64], [8, 64, 8, 8, 64, 64], [8, 8, 64, 64, 64, 8], [8, 8, 64, 64, 8, 64], [8, 8, 64, 8, 64, 64] and [8, 8, 8, 64, 64, 64]. %t A298672 Join[{1}, Table[SeriesCoefficient[Sum[x^k^3, {k, 1, n}]^n, {x, 0, n^3}], {n, 1, 23}]] %Y A298672 Cf. A000578, A030272, A232173, A259792, A290247, A291700, A298329, A298330, A298641, A298671. %K A298672 nonn,new %O A298672 0,7 %A A298672 _Ilya Gutkovskiy_, Jan 24 2018 %I A298671 %S A298671 1,1,2,3,4,5,146,4207,26329,257721,3556495,42685181,631230381, %T A298671 9409600499,142557084957,2781352245050,52598395446786,950288577530017, %U A298671 20768368026768594,448759012546543804,9652848877533217174,235179507693424886403,5756272592837812726164,140920987987840184113287 %N A298671 Number of ordered ways of writing n^3 as a sum of n nonnegative cubes. %H A298671 Index entries for sequences related to sums of cubes %F A298671 a(n) = [x^(n^3)] (Sum_{k>=0} x^(k^3))^n. %e A298671 a(3) = 3 because we have [27, 0, 0], [0, 27, 0] and [0, 0, 27]. %t A298671 Table[SeriesCoefficient[Sum[x^k^3, {k, 0, n}]^n, {x, 0, n^3}], {n, 0, 23}] %Y A298671 Cf. A000578, A030272, A232173, A259792, A290054, A290247, A291700, A298329, A298330, A298641, A298672. %K A298671 nonn,new %O A298671 0,3 %A A298671 _Ilya Gutkovskiy_, Jan 24 2018 %I A297554 %S A297554 1,4,7,12,19,28,51,76,115,204,307,460,819,1228,1843,3276,4915,7372, %T A297554 13107,19660,29491,52428,78643,117964,209715,314572,471859,838860, %U A297554 1258291,1887436,3355443,5033164,7549747,13421772,20132659,30198988,53687091,80530636 %N A297554 a(n) = a(n-2) + 4*a(n-3) - 4*a(n-5), where a(0) = 1, a(1) = 4, a(2) = 7, a(3) = 12, a(4) = 19, a(5) = 28. %C A297554 Conjecture: a(n) = least positive whose base-4 total variation is n; see A297551. %H A297554 Clark Kimberling, Table of n, a(n) for n = 0..1000 %H A297554 Index entries for linear recurrences with constant coefficients, signature (0,1,4,0,-4) %F A297554 G.f.: (1 + 4 x + 6 x^2 + 4 x^3 - 4 x^4 - 8 x^5)/(1 - x^2 - 4 x^3 + 4 x^5). %t A297554 Join[{1}, LinearRecurrence[{0, 1, 4, 0, -4}, {4, 7, 12, 19, 28}, 40]] %Y A297554 Cf. A297551. %K A297554 nonn,easy,new %O A297554 0,2 %A A297554 _Clark Kimberling_, Jan 21 2018 %I A297553 %S A297553 1,2,6,3,11,7,4,17,18,19,5,22,23,35,99,8,24,27,51,114,115,9,25,28,67, %T A297553 119,291,307,10,26,29,71,179,306,563,1587,12,33,30,75,275,311,819, %U A297553 1827,1843,13,38,31,76,290,371,1075,1842,4659,4915 %N A297553 Rectangular array R by antidiagonals: row n shows the positive integers whose base-4 digits have up-variation n, for n>=0. See Comments. %C A297553 Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See A297330 for a guide to related sequences and partitions of the natural numbers. %C A297553 Every positive integer occurs exactly once in the array, so that as a sequence this is a permutation of the positive integers. %C A297553 Conjecture: each column, after some number of initial terms, satisfies a homogeneous linear recurrence relation. %e A297553 Northwest corner: %e A297553 1 2 3 4 5 8 9 10 %e A297553 6 11 17 22 24 25 26 33 %e A297553 7 18 23 27 28 29 30 31 %e A297553 19 35 51 67 71 75 76 77 %e A297553 99 114 119 179 275 290 295 305 %e A297553 115 291 306 311 371 435 451 455 %e A297553 307 563 819 1075 1139 1203 1219 1223 %t A297553 g[n_, b_] := Differences[IntegerDigits[n, b]]; %t A297553 b = 4; z = 200000; u = Table[-Total[Select[g[n, b], # > 0 &]], {n, 1, z}] ; %t A297553 p[n_] := Position[u, n]; TableForm[Table[Take[Flatten[p[n]], 15], {n, 0, 9}]] %t A297553 v[n_, k_] := p[k - 1][[n]]; %t A297553 Table[v[k, n - k + 1], {n, 10}, {k, n, 1, -1}] // Flatten %Y A297553 Cf. A297556 (conjectured 1st column), A297551, A297552. %K A297553 nonn,tabl,easy,new %O A297553 1,2 %A A297553 _Clark Kimberling_, Jan 21 2018 %I A298592 %S A298592 1,2,1,8,5,3,50,34,25,16,432,307,243,189,125,4802,3506,2881,2401,1921, %T A298592 1296,65536,48729,40953,35328,30208,24583,16807,1062882,800738,683089, %U A298592 601441,531441,461441,379793,262144,20000000,15217031,13119879,11708091,10546875,9453125,8291909,6880121,4782969 %N A298592 Triangle read by rows: T(n,k) = number of parking functions of length n whose lead number is k. %H A298592 D. Foata and J. Riordan, Mappings of acyclic and parking functions, J. Aeq. Math., 10 (1974) 10-22. %F A298592 T(n,k) = Sum_{j=k..n} binomial(n-1, j-1)*j^(j-2)*(n+1-j)^(n-1-j). %F A298592 T(n,k) = A298593(n,k)/n. %F A298592 T(n,k) = Sum_{j=k..n} A298594(n,j). %F A298592 T(n,k) = (Sum_{j=k..n} A298597(n,j))/n. %F A298592 Sum_{j=1..n} T(n,k) = A000272(n+1). %e A298592 Triangle begins: %e A298592 1; %e A298592 2, 1; %e A298592 8, 5, 3; %e A298592 50, 34, 25, 16; %e A298592 432, 307, 243, 189, 125; %e A298592 4802, 3506, 2881, 2401, 1921, 1296; %e A298592 65536, 48729, 40953, 35328, 30208, 24583, 16807; %e A298592 1062882, 800738, 683089, 601441, 531441, 461441, 379793, 262144; %e A298592 ... %t A298592 Table[Sum[Binomial[n - 1, j - 1] j^(j - 2)*(n + 1 - j)^(n - 1 - j), {j, k, n}], {n, 9}, {k, n}] // Flatten (* _Michael De Vlieger_, Jan 22 2018 *) %Y A298592 Cf. A000272, A298593, A298594, A298597. %K A298592 easy,nonn,tabl,new %O A298592 1,2 %A A298592 _Rui Duarte_, Jan 22 2018 %I A298677 %S A298677 1,111,12209,1342879,147704481,16246150031,1786928798929, %T A298677 196545921732159,21618264461738561,2377812544869509551, %U A298677 261537761671184312049,28766775971285404815839,3164083819079723345430241,348020453322798282592510671,38279085781688731361830743569,4210351415532437651518789281919 %N A298677 a(n) = 110*a(n-1) - a(n-2) for n >= 2, a(0)=1, a(1)=111. %C A298677 Sequence {s_k(110)} of A269254. %C A298677 The sequence contains no primes; see A269254 for a proof by _N. J. A. Sloane_. %H A298677 Colin Barker, Table of n, a(n) for n = 0..450 %H A298677 Index entries for linear recurrences with constant coefficients, signature (110,-1). %F A298677 G.f.: (1 + x)/(1 - 110*x + x^2). %F A298677 a(n) = (1/18)*((55+12*sqrt(21))^(-n)*(9-2*sqrt(21) + (9+2*sqrt(21))*(55+12*sqrt(21))^(2*n))). - _Colin Barker_, Jan 25 2018 %t A298677 s[0, n_] := 1; s[1, n_] := n + 1; s[k_, n_] := n*s[k - 1, n] - s[k - 2, n]; Table[s[k, 110], {k, 0, 15}] %t A298677 LinearRecurrence[{110, -1}, {1, 111}, 15] %t A298677 CoefficientList[Series[(1 + x)/(1 - 110*x + x^2), {x, 0, 14}], x] %o A298677 (PARI) Vec((1 + x)/(1 - 110*x + x^2) + O(x^20)) \\ _Colin Barker_, Jan 25 2018 %Y A298677 Cf. A285992, A299107, A299109, A299110, A117522, A299100--A299102. %Y A298677 Cf. A269253, A269254, A294099, A298675, A298677, A298678, A299045, A299071. %K A298677 nonn,easy,new %O A298677 0,2 %A A298677 _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Jan 24 2018 %I A298675 %S A298675 1,2,-1,3,2,-2,4,7,2,-1,5,14,18,2,1,6,23,52,47,2,2,7,34,110,194,123,2, %T A298675 1,8,47,198,527,724,322,2,-1,9,62,322,1154,2525,2702,843,2,-2,10,79, %U A298675 488,2207,6726,12098,10084,2207,2,-1,11,98,702,3842,15127,39202,57965,37634,5778,2,1 %N A298675 Rectangular array A: first differences of row entries of array A294099, read by antidiagonals. %C A298675 From a problem in A269254. %F A298675 A(n,k) = T_k(n), n >= 1, k >= 1, where T_j(x) = x*T_{j-1}(x) - T_{j-2}(x), j >= 2, T_0(x) = 2, T_1(x) = x, (dilated Chebyshev polynomials of the first kind). %e A298675 Array begins: %e A298675 . 1 -1 -2 -1 1 2 1 -1 -2 -1 %e A298675 . 2 2 2 2 2 2 2 2 2 2 %e A298675 . 3 7 18 47 123 322 843 2207 5778 15127 %e A298675 . 4 14 52 194 724 2702 10084 37634 140452 524174 %e A298675 . 5 23 110 527 2525 12098 57965 277727 1330670 6375623 %e A298675 . 6 34 198 1154 6726 39202 228486 1331714 7761798 45239074 %e A298675 . 7 47 322 2207 15127 103682 710647 4870847 33385282 228826127 %e A298675 . 8 62 488 3842 30248 238142 1874888 14760962 116212808 914941502 %e A298675 . 9 79 702 6239 55449 492802 4379769 38925119 345946302 3074591599 %e A298675 . 10 98 970 9602 95050 940898 9313930 92198402 912670090 9034502498 %t A298675 t[n_, 0] := 2; t[n_, 1] := n; t[n_, k_] := n*t[n, k - 1] - t[n, k - 2]; Table[t[n, k], {n, 10}, {k, 10}] // Grid %Y A298675 Cf. A285992, A299107, A299109, A299110, A117522, A299100--A299102. %Y A298675 Cf. A269253, A269254, A294099, A298675, A298677, A298678, A299045, A299071. %K A298675 sign,tabl,new %O A298675 1,2 %A A298675 _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Jan 24 2018 %I A298731 %S A298731 1,0,0,1,0,0,1,0,0,1,1,0,1,1,0,1,1,0,1,1,1,1,0,1,1,1,1,0,1,0,2,1,0,2, %T A298731 0,1,2,0,1,1,2,0,1,1,0,4,1,0,2,1,1,2,1,0,3,2,1,2,1,1,3,2,0,3,2,1,4,1, %U A298731 1,3,2,1,3,2,1,4,2,1,3,2,1,3,2,1,4,2,1,3,1,1,4,2,1,4,2,0,4,1,1,4,2,1,3,3,0,4,1 %N A298731 Number of distinct representations of n as a sum of four terms of A020330 (including 0), where order does not matter. %H A298731 Parthasarathy Madhusudan, Dirk Nowotka, Aayush Rajasekaran, Jeffrey Shallit, Lagrange's Theorem for Binary Squares, arXiv:1710.04247 [math.NT], Oct 11 2017. %e A298731 For n = 45, the a(45) = 4 solutions are 45 = 15+15+15 = 36+3+3+3 = 15+10+10+10. %Y A298731 Cf. A290335, which is the same sequence where order matters. %K A298731 nonn,new %O A298731 0,31 %A A298731 _Jeffrey Shallit_, Jan 25 2018 %I A298487 %S A298487 1,10,19,43,67,83,92,293,691,958,7849,49670,94976,880096,7090761, %T A298487 80890670,798992994,9999069559,808009099075,8979948969844, %U A298487 898989690790838 %N A298487 a(n) is the least number with persistence n as defined using A114570. %C A298487 Repeat A114570 until one digit remains. %C A298487 a(n) is the first number with persistence n in base 10. %C A298487 a(n+1) <= Sum_{i=0..a(n)-1} 10^i showing the sequence is infinite. %C A298487 a(n) does not necessarily pass through a(n-1) on the first step. %e A298487 a(5) = 83 because: %e A298487 83 -> 8^2 + 3^1 = 67; %e A298487 67 -> 6^2 + 7^1 = 43; %e A298487 43 -> 4^2 + 3^1 = 19; %e A298487 17 -> 1^2 + 9^1 = 10; %e A298487 10 -> 1^2 + 0^1 = 1; %e A298487 83 is the least integer to take 5 steps to get to 1 digit. %o A298487 (PARI) a114570(n) = my(d=digits(n), k=#d); sum(i=1, k, d[i]^(k+1-i)); %o A298487 p(n) = my(ip=0); while(n >= 10, n = a114570(n); ip++); ip; %o A298487 a(n) = {my(k=1); while (p(k) != n, k++); k;} \\ _Michel Marcus_, Jan 25 2018 %Y A298487 Cf. A114570. %K A298487 nonn,base,more,new %O A298487 0,2 %A A298487 _John Harmon_, Jan 20 2018 %E A298487 a(19) from _Giovanni Resta_, Jan 22 2018 %E A298487 a(20) from _Giovanni Resta_, Feb 01 2018 %I A298684 %S A298684 60,540,660,1200,1320,1620,2160,3060,5580,6120,6600,6720,8100,9180, %T A298684 9240,9600,9720,9900,11160,12240,12300,12600,13200,13440,13680,15120, %U A298684 15360,18300,18480,19440,19800,21000,22500,24480,24840,26880,27360,28920,29400,30240,30780 %N A298684 Numbers i such that Fibonacci(i) is divisible by i, i+1, and i+2. %C A298684 A subsequence of A217738. %H A298684 Chai Wah Wu, Table of n, a(n) for n = 1..1000 %t A298684 fQ[n_] := Mod[ Fibonacci@ n, {n, n +1, n +2}] == {0, 0, 0}; Select[60 Range@513, fQ] (* _Robert G. Wilson v_, Jan 26 2018 *) %o A298684 (Python) %o A298684 from __future__ import division %o A298684 A298684_list, n, a, b = [], 1, 1, 1 %o A298684 while len(A298684_list) < 1000: %o A298684 if not (a % (n*(n+1)*(n+2)//(1 if n % 2 else 2))): %o A298684 A298684_list.append(n) %o A298684 n += 1 %o A298684 a, b = b, a+b # _Chai Wah Wu_, Jan 26 2018 %o A298684 (MAGMA) [n: n in [2..10^5] | IsZero(Fibonacci(n) mod (n)) and IsZero(Fibonacci(n) mod (n+1)) and IsZero(Fibonacci(n) mod (n+2))]; // _Vincenzo Librandi_, Jan 27 2018 %Y A298684 Cf. A000045, A023172, A217738, A221018, A225219. %K A298684 nonn,new %O A298684 1,1 %A A298684 _Alex Ratushnyak_, Jan 24 2018 %I A298727 %S A298727 1,1,1,1,5,1,1,7,7,1,1,18,5,18,1,1,31,15,15,31,1,1,65,21,34,21,65,1,1, %T A298727 130,57,77,77,57,130,1,1,253,119,230,336,230,119,253,1,1,519,285,712, %U A298727 1041,1041,712,285,519,1,1,1018,725,2167,3676,4001,3676,2167,725,1018,1 %N A298727 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 2, 3, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298727 Table starts %C A298727 .1...1...1....1.....1......1.......1........1........1.........1..........1 %C A298727 .1...5...7...18....31.....65.....130......253......519......1018.......2055 %C A298727 .1...7...5...15....21.....57.....119......285......725......1833.......4807 %C A298727 .1..18..15...34....77....230.....712.....2167.....6694.....20775......64197 %C A298727 .1..31..21...77...336...1041....3676....12970....45311....162322.....581460 %C A298727 .1..65..57..230..1041...4001...16320....67159...281930...1196239....5097819 %C A298727 .1.130.119..712..3676..16320...76983...380106..1892587...9597169...48598454 %C A298727 .1.253.285.2167.12970..67159..380106..2232433.13202628..79480110..477560387 %C A298727 .1.519.725.6694.45311.281930.1892587.13202628.92349689.659941442.4710349603 %H A298727 R. H. Hardin, Table of n, a(n) for n = 1..220 %F A298727 Empirical for column k: %F A298727 k=1: a(n) = a(n-1) %F A298727 k=2: a(n) = a(n-1) +3*a(n-2) -4*a(n-4) for n>5 %F A298727 k=3: [order 17] for n>18 %F A298727 k=4: [order 58] for n>60 %e A298727 Some solutions for n=5 k=4 %e A298727 ..0..0..1..1. .0..1..0..1. .0..0..1..1. .0..0..0..0. .0..0..0..0 %e A298727 ..1..0..0..1. .0..0..0..0. .0..1..0..1. .0..1..1..0. .0..0..0..0 %e A298727 ..1..1..1..0. .1..0..1..0. .1..1..0..1. .0..1..0..1. .0..0..0..0 %e A298727 ..0..0..0..0. .1..1..1..0. .0..1..0..1. .1..0..0..1. .0..0..0..0 %e A298727 ..1..0..1..0. .0..1..0..0. .0..0..1..1. .1..1..1..1. .0..0..0..0 %Y A298727 Column 2 is A297937. %K A298727 nonn,tabl,new %O A298727 1,5 %A A298727 _R. H. Hardin_, Jan 25 2018 %I A298726 %S A298726 1,130,119,712,3676,16320,76983,380106,1892587,9597169,48598454, %T A298726 246246047,1246714439,6319048613,32032228345,162468899792, %U A298726 824193865328,4181860443550,21218926355643,107671599941726,546370612376581 %N A298726 Number of nX7 0..1 arrays with every element equal to 0, 2, 3, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298726 Column 7 of A298727. %H A298726 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298726 Some solutions for n=7 %e A298726 ..0..0..0..0..1..1..1. .0..0..1..1..1..0..0. .0..0..0..1..1..1..1 %e A298726 ..0..1..1..0..1..1..1. .0..1..0..0..0..1..0. .0..0..0..1..0..0..1 %e A298726 ..1..0..1..0..1..1..1. .0..1..1..1..1..1..0. .0..0..0..1..0..1..0 %e A298726 ..1..0..1..0..0..0..0. .1..0..0..0..0..0..0. .0..0..0..1..1..0..0 %e A298726 ..0..1..0..1..1..1..0. .1..1..0..1..1..1..1. .0..0..0..1..0..1..0 %e A298726 ..1..0..0..0..1..0..1. .0..0..0..1..1..1..1. .0..0..0..1..0..1..0 %e A298726 ..1..1..1..1..0..1..1. .0..1..0..1..1..1..1. .0..0..0..1..1..0..0 %Y A298726 Cf. A298727. %K A298726 nonn,new %O A298726 1,2 %A A298726 _R. H. Hardin_, Jan 25 2018 %I A298725 %S A298725 1,65,57,230,1041,4001,16320,67159,281930,1196239,5097819,21700708, %T A298725 92361718,393375679,1675409096,7138020130,30417861481,129627100074, %U A298725 552440958797,2354439976293,10034502784748,42766999408555 %N A298725 Number of nX6 0..1 arrays with every element equal to 0, 2, 3, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298725 Column 6 of A298727. %H A298725 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298725 Some solutions for n=7 %e A298725 ..0..1..0..1..1..1. .0..1..0..1..0..0. .0..0..0..0..0..0. .0..0..0..1..0..1 %e A298725 ..0..0..0..1..1..1. .1..1..1..1..0..1. .0..1..1..1..1..0. .0..0..0..1..1..1 %e A298725 ..1..1..0..1..1..1. .1..0..1..0..0..0. .0..1..1..1..1..0. .0..0..0..1..0..0 %e A298725 ..1..0..0..0..0..0. .0..0..0..0..0..1. .0..1..1..1..1..0. .1..1..1..1..1..0 %e A298725 ..0..1..1..1..1..0. .1..1..1..0..1..1. .0..0..0..0..0..0. .1..0..0..0..0..1 %e A298725 ..0..0..1..0..0..1. .1..1..1..0..0..0. .1..1..1..0..1..0. .1..0..1..1..1..0 %e A298725 ..0..1..1..0..1..1. .1..1..1..0..1..0. .0..1..0..0..1..1. .1..1..0..0..0..0 %Y A298725 Cf. A298727. %K A298725 nonn,new %O A298725 1,2 %A A298725 _R. H. Hardin_, Jan 25 2018 %I A298724 %S A298724 1,31,21,77,336,1041,3676,12970,45311,162322,581460,2074761,7425880, %T A298724 26606520,95200209,340938956,1221189411,4373317686,15664416165, %U A298724 56107953172,200965985895,719842577248,2578420648657,9235682972244 %N A298724 Number of nX5 0..1 arrays with every element equal to 0, 2, 3, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298724 Column 5 of A298727. %H A298724 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298724 Some solutions for n=5 %e A298724 ..0..1..0..1..1. .0..0..0..0..0. .0..0..0..0..0. .0..0..1..0..0 %e A298724 ..0..0..0..0..1. .0..0..0..0..0. .1..0..1..1..0. .0..1..1..0..1 %e A298724 ..1..0..1..0..1. .1..1..1..1..1. .0..0..1..0..1. .0..1..0..0..0 %e A298724 ..1..1..1..1..1. .1..1..1..1..1. .1..0..1..0..1. .0..0..1..1..1 %e A298724 ..0..1..0..1..0. .1..1..1..1..1. .0..0..0..1..1. .1..0..0..1..0 %Y A298724 Cf. A298727. %K A298724 nonn,new %O A298724 1,2 %A A298724 _R. H. Hardin_, Jan 25 2018 %I A298723 %S A298723 1,18,15,34,77,230,712,2167,6694,20775,64197,200062,624696,1949420, %T A298723 6084096,19000957,59343184,185364275,579116320,1809395723,5653433396, %U A298723 17664869660,55197920461,172481319644,538973897693,1684218210917 %N A298723 Number of nX4 0..1 arrays with every element equal to 0, 2, 3, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298723 Column 4 of A298727. %H A298723 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298723 Empirical: a(n) = 6*a(n-1) -9*a(n-2) -a(n-3) +2*a(n-4) -2*a(n-5) +15*a(n-6) +26*a(n-7) -157*a(n-8) +466*a(n-9) -220*a(n-10) -444*a(n-11) +96*a(n-12) -157*a(n-13) -652*a(n-14) +146*a(n-15) +1984*a(n-16) +130*a(n-17) +2025*a(n-18) -2473*a(n-19) +6365*a(n-20) -4168*a(n-21) -11764*a(n-22) +5390*a(n-23) -9469*a(n-24) +8003*a(n-25) -14173*a(n-26) +26192*a(n-27) -10436*a(n-28) +27101*a(n-29) -15251*a(n-30) +9559*a(n-31) +14483*a(n-32) -50379*a(n-33) +34809*a(n-34) -58786*a(n-35) +60997*a(n-36) -73488*a(n-37) +83309*a(n-38) -55321*a(n-39) +43987*a(n-40) -38228*a(n-41) +37989*a(n-42) -14291*a(n-43) +2903*a(n-44) -12735*a(n-45) +11717*a(n-46) -10534*a(n-47) +5177*a(n-48) +1621*a(n-49) +1470*a(n-50) -2283*a(n-51) -323*a(n-52) -752*a(n-53) +511*a(n-54) +22*a(n-55) +60*a(n-56) -80*a(n-57) +16*a(n-58) for n>60 %e A298723 Some solutions for n=5 %e A298723 ..0..0..0..0. .0..0..0..0. .0..1..1..0. .0..0..1..0. .0..0..0..0 %e A298723 ..0..1..1..0. .0..1..1..0. .0..0..1..1. .0..1..1..1. .0..0..0..0 %e A298723 ..1..0..1..0. .0..1..0..1. .1..1..0..1. .1..0..0..0. .0..0..0..0 %e A298723 ..1..0..0..1. .1..0..0..1. .0..1..0..0. .0..1..1..0. .1..1..1..1 %e A298723 ..1..1..1..1. .1..1..1..1. .1..1..1..0. .0..0..1..1. .1..1..1..1 %Y A298723 Cf. A298727. %K A298723 nonn,new %O A298723 1,2 %A A298723 _R. H. Hardin_, Jan 25 2018 %I A298722 %S A298722 1,7,5,15,21,57,119,285,725,1833,4807,12843,34439,93327,254085,693267, %T A298722 1896489,5194309,14237415,39049277,107136761,294009425,806965323, %U A298722 2215076227,6080649203,16692824711,45826947389,125811287255,345400728813 %N A298722 Number of nX3 0..1 arrays with every element equal to 0, 2, 3, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298722 Column 3 of A298727. %H A298722 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298722 Empirical: a(n) = 5*a(n-1) -5*a(n-2) -5*a(n-3) -2*a(n-4) +18*a(n-5) +14*a(n-6) -30*a(n-7) -8*a(n-8) -22*a(n-9) +4*a(n-10) +84*a(n-11) +3*a(n-12) -33*a(n-13) -9*a(n-14) -21*a(n-15) +4*a(n-16) +4*a(n-17) for n>18 %e A298722 Some solutions for n=5 %e A298722 ..0..1..1. .0..0..1. .0..1..0. .0..0..0. .0..0..0. .0..0..1. .0..1..1 %e A298722 ..0..0..1. .1..0..0. .1..1..1. .0..0..0. .0..0..0. .1..0..0. .1..1..0 %e A298722 ..1..0..1. .1..1..1. .0..0..0. .0..0..0. .0..0..0. .0..0..1. .0..0..0 %e A298722 ..1..1..1. .1..0..1. .0..0..0. .1..1..1. .0..0..0. .1..0..0. .0..1..0 %e A298722 ..0..1..0. .0..0..0. .0..0..0. .0..1..0. .0..0..0. .0..0..1. .1..1..1 %Y A298722 Cf. A298727. %K A298722 nonn,new %O A298722 1,2 %A A298722 _R. H. Hardin_, Jan 25 2018 %I A298721 %S A298721 1,5,5,34,336,4001,76983,2232433,92349689,5607760208 %N A298721 Number of nXn 0..1 arrays with every element equal to 0, 2, 3, 5 or 8 king-move adjacent elements, with upper left element zero. %C A298721 Diagonal of A298727. %e A298721 Some solutions for n=5 %e A298721 ..0..1..0..1..1. .0..0..1..1..1. .0..0..0..0..0. .0..1..0..1..0 %e A298721 ..0..0..0..0..1. .0..1..0..1..0. .0..0..0..0..0. .1..1..1..1..1 %e A298721 ..0..1..1..1..0. .1..1..0..1..1. .0..0..0..0..0. .1..0..1..0..1 %e A298721 ..1..0..0..0..1. .0..1..0..1..0. .1..1..1..1..1. .0..0..0..0..0 %e A298721 ..1..1..0..1..1. .0..0..1..0..0. .1..1..1..1..1. .0..1..0..1..0 %Y A298721 Cf. A298727. %K A298721 nonn,new %O A298721 1,2 %A A298721 _R. H. Hardin_, Jan 25 2018 %I A298719 %S A298719 0,0,0,0,1,0,0,2,2,0,0,5,4,5,0,0,10,13,13,10,0,0,25,63,59,63,25,0,0, %T A298719 54,264,346,346,264,54,0,0,125,1005,2246,3508,2246,1005,125,0,0,282, %U A298719 4113,13650,34704,34704,13650,4113,282,0,0,641,16720,87117,309593,563977 %N A298719 T(n,k)=Number of nXk 0..1 arrays with every element equal to 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298719 Table starts %C A298719 .0...0.....0......0........0..........0............0.............0 %C A298719 .0...1.....2......5.......10.........25...........54...........125 %C A298719 .0...2.....4.....13.......63........264.........1005..........4113 %C A298719 .0...5....13.....59......346.......2246........13650.........87117 %C A298719 .0..10....63....346.....3508......34704.......309593.......2960634 %C A298719 .0..25...264...2246....34704.....563977......8287084.....125919136 %C A298719 .0..54..1005..13650...309593....8287084....193439202....4672566734 %C A298719 .0.125..4113..87117..2960634..125919136...4672566734..177476979652 %C A298719 .0.282.16720.550582.27934888.1909040369.112689610775.6748480839625 %H A298719 R. H. Hardin, Table of n, a(n) for n = 1..180 %F A298719 Empirical for column k: %F A298719 k=1: a(n) = a(n-1) %F A298719 k=2: a(n) = 3*a(n-2) +4*a(n-3) +2*a(n-4) %F A298719 k=3: [order 18] %F A298719 k=4: [order 63] for n>64 %e A298719 Some solutions for n=5 k=4 %e A298719 ..0..0..1..1. .0..0..1..1. .0..0..1..1. .0..0..0..1. .0..1..1..1 %e A298719 ..0..1..0..1. .0..1..0..1. .0..1..0..1. .0..0..1..1. .0..0..1..1 %e A298719 ..1..0..0..0. .0..1..0..0. .1..1..0..0. .0..1..0..1. .0..1..0..1 %e A298719 ..0..1..1..0. .1..0..0..1. .1..0..0..1. .1..0..1..1. .1..0..0..1 %e A298719 ..0..0..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1 %Y A298719 Column 2 is A297860. %K A298719 nonn,tabl,new %O A298719 1,8 %A A298719 _R. H. Hardin_, Jan 25 2018 %I A298718 %S A298718 0,54,1005,13650,309593,8287084,193439202,4672566734,112689610775, %T A298718 2731899175896,66122998854098,1599672498018255,38731486336940251, %U A298718 937609098603118557,22696092440623817595,549420999860489552558 %N A298718 Number of nX7 0..1 arrays with every element equal to 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298718 Column 7 of A298719. %H A298718 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298718 Some solutions for n=5 %e A298718 ..0..0..1..0..0..1..1. .0..0..0..1..1..1..1. .0..1..1..1..1..0..0 %e A298718 ..0..1..0..1..0..0..1. .0..0..1..0..0..0..1. .0..0..1..1..0..0..0 %e A298718 ..0..1..0..1..0..1..0. .0..1..0..1..1..0..0. .0..1..1..1..1..1..1 %e A298718 ..1..0..0..0..1..0..0. .1..0..0..1..1..0..1. .0..1..1..0..1..0..1 %e A298718 ..1..1..1..1..0..0..0. .1..1..1..0..0..1..1. .0..0..0..0..0..0..0 %Y A298718 Cf. A298719. %K A298718 nonn,new %O A298718 1,2 %A A298718 _R. H. Hardin_, Jan 25 2018 %I A298717 %S A298717 0,25,264,2246,34704,563977,8287084,125919136,1909040369,29074548954, %T A298717 442306308211,6726735527836,102369187396017,1557632901882843, %U A298717 23699769940163807,360615767570355940,5487096721395970064 %N A298717 Number of nX6 0..1 arrays with every element equal to 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298717 Column 6 of A298719. %H A298717 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298717 Some solutions for n=5 %e A298717 ..0..0..1..0..1..1. .0..0..0..0..1..1. .0..0..1..1..0..0. .0..0..1..1..1..0 %e A298717 ..0..1..0..1..0..1. .0..1..1..0..1..1. .0..1..1..1..1..0. .0..0..1..1..0..0 %e A298717 ..1..1..1..1..0..0. .0..1..1..1..0..0. .0..1..1..1..1..0. .0..1..1..1..1..0 %e A298717 ..0..0..1..1..0..1. .0..0..1..1..1..0. .0..0..1..1..0..0. .0..1..1..1..0..0 %e A298717 ..0..1..1..1..1..1. .0..1..1..1..0..0. .0..0..1..1..0..0. .0..0..1..1..1..0 %Y A298717 Cf. A298719. %K A298717 nonn,new %O A298717 1,2 %A A298717 _R. H. Hardin_, Jan 25 2018 %I A298716 %S A298716 0,10,63,346,3508,34704,309593,2960634,27934888,265454828,2518958779, %T A298716 23904132070,227038517931,2155865905516,20471559784593, %U A298716 194404183107602,1846100569583663,17531009460252304,166478738684389946 %N A298716 Number of nX5 0..1 arrays with every element equal to 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298716 Column 5 of A298719. %H A298716 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298716 Some solutions for n=5 %e A298716 ..0..0..1..0..0. .0..0..0..0..0. .0..0..1..0..0. .0..0..0..0..0 %e A298716 ..0..1..0..1..0. .0..1..1..1..0. .0..1..1..1..0. .0..1..0..1..0 %e A298716 ..0..1..1..1..0. .1..1..1..0..0. .0..1..1..1..0. .0..1..1..1..0 %e A298716 ..0..1..1..1..0. .0..1..1..1..0. .0..1..0..1..0. .0..1..1..1..0 %e A298716 ..0..0..1..0..0. .0..0..0..0..0. .0..0..1..0..0. .0..0..1..0..0 %Y A298716 Cf. A298719. %K A298716 nonn,new %O A298716 1,2 %A A298716 _R. H. Hardin_, Jan 25 2018 %I A298715 %S A298715 0,5,13,59,346,2246,13650,87117,550582,3489783,22151146,140554255, %T A298715 891978232,5661362187,35930336202,228040023568,1447314125762, %U A298715 9185725194376,58299474631379,370012068242202,2348373088230451 %N A298715 Number of nX4 0..1 arrays with every element equal to 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298715 Column 4 of A298719. %H A298715 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298715 Empirical: a(n) = 3*a(n-1) +21*a(n-2) +26*a(n-3) -83*a(n-4) -381*a(n-5) -615*a(n-6) +35*a(n-7) +3867*a(n-8) +7013*a(n-9) -10556*a(n-10) -28387*a(n-11) +36768*a(n-12) +76311*a(n-13) -45835*a(n-14) +3816*a(n-15) +97956*a(n-16) -117608*a(n-17) -52644*a(n-18) -747689*a(n-19) -1709818*a(n-20) -645060*a(n-21) -356415*a(n-22) +1628581*a(n-23) +4221676*a(n-24) +2300743*a(n-25) +1249904*a(n-26) +3932427*a(n-27) +17279908*a(n-28) +9184386*a(n-29) -17055518*a(n-30) +1579068*a(n-31) +5489858*a(n-32) -25487233*a(n-33) -17021794*a(n-34) -4854033*a(n-35) -41985127*a(n-36) -27002556*a(n-37) -21698789*a(n-38) -3750642*a(n-39) -47747959*a(n-40) -17666861*a(n-41) +21595450*a(n-42) -9559888*a(n-43) -13055872*a(n-44) -8853708*a(n-45) +8773297*a(n-46) +25433464*a(n-47) -139020*a(n-48) -17186112*a(n-49) +22110645*a(n-50) -10615240*a(n-51) +6879505*a(n-52) -2098646*a(n-53) +217015*a(n-54) -319802*a(n-55) -196014*a(n-56) -101709*a(n-57) +40087*a(n-58) +18226*a(n-59) -14278*a(n-60) +4444*a(n-61) -144*a(n-62) -768*a(n-63) for n>64 %e A298715 Some solutions for n=7 %e A298715 ..0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..0..0. .0..0..1..1 %e A298715 ..0..1..0..1. .0..1..1..0. .0..0..1..0. .0..1..1..0. .0..0..0..1 %e A298715 ..1..1..1..1. .1..0..0..0. .1..1..1..1. .1..0..1..0. .0..1..0..1 %e A298715 ..1..1..1..1. .1..1..1..0. .1..1..1..1. .1..0..0..0. .1..1..1..1 %e A298715 ..1..0..0..1. .1..1..1..0. .1..0..1..0. .1..0..0..0. .1..1..1..1 %e A298715 ..0..0..1..1. .1..0..1..0. .1..0..0..0. .1..0..1..1. .1..0..0..1 %e A298715 ..0..0..0..1. .0..0..0..0. .1..1..0..0. .1..1..1..1. .0..0..0..0 %Y A298715 Cf. A298719. %K A298715 nonn,new %O A298715 1,2 %A A298715 _R. H. Hardin_, Jan 25 2018 %I A298714 %S A298714 0,2,4,13,63,264,1005,4113,16720,67541,273173,1105028,4470455, %T A298714 18086203,73164414,295979201,1197372779,4843899390,19595676337, %U A298714 79273069299,320694228442,1297348415225,5248341466951,21231835277428,85892055453779 %N A298714 Number of nX3 0..1 arrays with every element equal to 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298714 Column 3 of A298719. %H A298714 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298714 Empirical: a(n) = 3*a(n-1) +4*a(n-2) +3*a(n-3) -2*a(n-4) -18*a(n-5) -29*a(n-6) -32*a(n-7) +74*a(n-8) +74*a(n-9) -90*a(n-10) -73*a(n-11) +33*a(n-12) -21*a(n-13) -29*a(n-14) +32*a(n-15) +4*a(n-16) +4*a(n-17) +8*a(n-18) %e A298714 Some solutions for n=7 %e A298714 ..0..0..1. .0..0..1. .0..1..1. .0..0..0. .0..0..0. .0..0..0. .0..0..0 %e A298714 ..0..1..1. .0..1..1. .0..0..1. .1..0..0. .0..1..0. .1..0..1. .1..0..1 %e A298714 ..0..0..0. .0..0..0. .1..1..1. .1..1..1. .1..1..1. .1..1..1. .1..1..1 %e A298714 ..0..0..0. .0..0..0. .1..1..1. .1..1..1. .1..1..1. .1..1..1. .1..1..1 %e A298714 ..0..1..0. .0..1..0. .1..0..1. .1..0..1. .1..0..1. .1..0..0. .0..0..1 %e A298714 ..1..0..1. .1..1..0. .1..0..0. .1..0..0. .1..0..0. .0..0..1. .0..1..1 %e A298714 ..1..1..1. .1..0..0. .1..1..0. .1..1..0. .1..1..0. .0..1..1. .0..0..1 %Y A298714 Cf. A298719. %K A298714 nonn,new %O A298714 1,2 %A A298714 _R. H. Hardin_, Jan 25 2018 %I A298713 %S A298713 0,1,4,59,3508,563977,193439202,177476979652,405759343397572 %N A298713 Number of nXn 0..1 arrays with every element equal to 2, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298713 Diagonal of A298719. %e A298713 Some solutions for n=5 %e A298713 ..0..0..0..0..0. .0..0..1..0..0. .0..0..0..0..0. .0..0..0..0..0 %e A298713 ..0..1..1..1..0. .0..1..1..1..0. .0..1..0..1..0. .0..1..1..1..0 %e A298713 ..1..1..1..0..0. .0..1..1..1..0. .0..1..1..1..0. .1..0..1..1..1 %e A298713 ..0..1..1..1..0. .0..1..0..1..0. .0..1..1..1..0. .0..1..1..1..0 %e A298713 ..0..0..0..0..0. .0..0..0..0..0. .0..0..1..0..0. .0..0..0..0..0 %Y A298713 Cf. A298719. %K A298713 nonn,new %O A298713 1,3 %A A298713 _R. H. Hardin_, Jan 25 2018 %I A298712 %S A298712 0,1,1,0,4,0,1,3,3,1,0,13,0,13,0,1,32,2,2,32,1,0,53,6,11,6,53,0,1,125, %T A298712 20,32,32,20,125,1,0,386,22,87,781,87,22,386,0,1,727,92,110,1354,1354, %U A298712 110,92,727,1,0,1601,206,385,2409,3950,2409,385,206,1601,0,1,4568,460,908 %N A298712 T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298712 Table starts %C A298712 .0...1...0...1......0......1.......0........1.........0..........1...........0 %C A298712 .1...4...3..13.....32.....53.....125......386.......727.......1601........4568 %C A298712 .0...3...0...2......6.....20......22.......92.......206........460........1176 %C A298712 .1..13...2..11.....32.....87.....110......385.......908.......2760........6454 %C A298712 .0..32...6..32....781...1354....2409....34250....109459.....294045.....2120905 %C A298712 .1..53..20..87...1354...3950....9586....92375....384519....1468344.....9531019 %C A298712 .0.125..22.110...2409...9586...17377...241249...1201122....6055242....45702804 %C A298712 .1.386..92.385..34250..92375..241249..8695221..49691754..274634548..4587455345 %C A298712 .0.727.206.908.109459.384519.1201122.49691754.359275256.2514115870.49068141912 %H A298712 R. H. Hardin, Table of n, a(n) for n = 1..199 %F A298712 Empirical for column k: %F A298712 k=1: a(n) = a(n-2) %F A298712 k=2: a(n) = a(n-1) +a(n-2) +8*a(n-3) -16*a(n-5) %F A298712 k=3: [order 17] for n>18 %F A298712 k=4: [order 70] %e A298712 Some solutions for n=5 k=4 %e A298712 ..0..1..0..1. .0..0..1..0. .0..0..1..1. .0..1..1..1. .0..1..1..1 %e A298712 ..0..1..0..1. .0..0..1..0. .0..0..0..0. .1..0..1..1. .1..0..1..1 %e A298712 ..1..1..1..1. .1..1..1..1. .1..1..1..1. .0..0..0..0. .0..0..0..0 %e A298712 ..0..1..0..1. .0..0..1..0. .0..0..1..1. .1..0..1..1. .0..1..1..1 %e A298712 ..0..1..0..1. .0..0..0..1. .0..0..0..0. .1..0..1..1. .1..0..1..1 %Y A298712 Column 2 is A298057. %K A298712 nonn,tabl,new %O A298712 1,5 %A A298712 _R. H. Hardin_, Jan 25 2018 %I A298711 %S A298711 0,125,22,110,2409,9586,17377,241249,1201122,6055242,45702804, %T A298711 299335380,1843128506,13177259868,89426659750,604369128394, %U A298711 4250651593956,29338538378352,203129511477403,1429210885229805,9929275442796545 %N A298711 Number of nX7 0..1 arrays with every element equal to 1, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298711 Column 7 of A298712. %H A298711 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298711 Some solutions for n=5 %e A298711 ..0..1..1..1..0..1..1. .0..0..1..1..0..0..1. .0..1..0..1..1..0..0 %e A298711 ..1..0..1..1..0..1..1. .1..1..1..1..0..0..1. .0..1..0..1..1..0..0 %e A298711 ..1..1..1..0..0..0..0. .1..1..1..1..1..1..1. .1..1..1..1..1..1..1 %e A298711 ..1..1..1..1..0..1..1. .0..0..1..1..0..0..1. .0..1..0..1..1..1..1 %e A298711 ..0..0..1..1..0..1..1. .1..1..1..1..0..0..1. .0..1..0..1..1..0..0 %Y A298711 Cf. A298712. %K A298711 nonn,new %O A298711 1,2 %A A298711 _R. H. Hardin_, Jan 25 2018 %I A298710 %S A298710 1,53,20,87,1354,3950,9586,92375,384519,1468344,9531019,48632741, %T A298710 222839718,1283757387,6897783261,35029987304,193742947555, %U A298710 1057480722066,5617597365512,30811375701229,168849070826644,914727570968224 %N A298710 Number of nX6 0..1 arrays with every element equal to 1, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298710 Column 6 of A298712. %H A298710 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298710 Some solutions for n=5 %e A298710 ..0..1..1..1..0..1. .0..1..0..0..0..1. .0..1..0..0..1..0. .0..1..0..1..1..0 %e A298710 ..1..0..1..1..0..1. .0..1..0..0..1..0. .1..0..0..0..1..0. .0..1..0..1..1..0 %e A298710 ..0..0..1..1..1..1. .1..1..1..1..1..1. .1..1..0..0..0..0. .0..0..0..0..0..0 %e A298710 ..0..1..1..1..1..0. .0..1..0..0..1..0. .1..1..0..0..1..0. .0..0..1..1..0..1 %e A298710 ..1..0..1..1..0..1. .0..1..0..0..0..1. .0..0..0..0..0..1. .1..1..1..1..1..0 %Y A298710 Cf. A298712. %K A298710 nonn,new %O A298710 1,2 %A A298710 _R. H. Hardin_, Jan 25 2018 %I A298709 %S A298709 0,32,6,32,781,1354,2409,34250,109459,294045,2120905,9245389,30828598, %T A298709 168214873,796498729,3061252232,14764004332,70505421866,293384815488, %U A298709 1348082131155,6351782335450,27606509717199,124692900069265 %N A298709 Number of nX5 0..1 arrays with every element equal to 1, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298709 Column 5 of A298712. %H A298709 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298709 Some solutions for n=5 %e A298709 ..0..1..0..1..0. .0..1..0..0..1. .0..0..1..0..1. .0..1..0..1..0 %e A298709 ..1..0..0..0..1. .1..0..0..1..0. .1..1..1..1..0. .0..1..0..1..0 %e A298709 ..0..0..0..0..0. .1..1..0..1..1. .1..1..1..1..1. .1..1..1..1..1 %e A298709 ..1..0..0..1..0. .0..1..0..1..0. .0..1..0..1..1. .0..1..0..1..0 %e A298709 ..0..1..0..0..1. .1..0..0..0..1. .1..0..0..0..0. .1..0..0..0..1 %Y A298709 Cf. A298712. %K A298709 nonn,new %O A298709 1,2 %A A298709 _R. H. Hardin_, Jan 25 2018 %I A298708 %S A298708 1,13,2,11,32,87,110,385,908,2760,6454,17925,51777,141793,378486, %T A298708 1095621,3060776,8412703,23884402,67448745,187193308,529617392, %U A298708 1496665754,4179115507,11797048913,33344442453,93407406180,263365123043 %N A298708 Number of nX4 0..1 arrays with every element equal to 1, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298708 Column 4 of A298712. %H A298708 R. H. Hardin, Table of n, a(n) for n = 1..210 %H A298708 R. H. Hardin, Empirical recurrence of order 70 %F A298708 Empirical recurrence of order 70 (see link above) %e A298708 Some solutions for n=5 %e A298708 ..0..1..0..0. .0..0..0..0. .0..0..1..1. .0..1..0..0. .0..0..1..0 %e A298708 ..0..1..0..0. .0..0..1..1. .0..0..1..1. .0..1..0..0. .0..0..1..0 %e A298708 ..1..1..1..0. .1..1..1..1. .1..1..0..0. .1..1..1..1. .0..1..1..1 %e A298708 ..0..1..0..0. .0..0..0..0. .0..0..1..1. .0..1..0..0. .0..0..1..0 %e A298708 ..0..1..0..0. .0..0..1..1. .0..0..1..1. .0..1..0..0. .0..0..1..0 %Y A298708 Cf. A298712. %K A298708 nonn,new %O A298708 1,2 %A A298708 _R. H. Hardin_, Jan 25 2018 %I A298707 %S A298707 0,3,0,2,6,20,22,92,206,460,1176,2966,6898,17232,42850,103328,254344, %T A298707 630010,1536786,3773848,9309476,22822020,56028990,137918372,338696066, %U A298707 831789610,2045396676,5025784292,12346278992,30346864108,74575773746 %N A298707 Number of nX3 0..1 arrays with every element equal to 1, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298707 Column 3 of A298712. %H A298707 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298707 Empirical: a(n) = a(n-1) +2*a(n-2) +12*a(n-3) -5*a(n-4) -23*a(n-5) -50*a(n-6) -2*a(n-7) +70*a(n-8) +84*a(n-9) +33*a(n-10) -86*a(n-11) -24*a(n-12) +2*a(n-13) +44*a(n-14) -14*a(n-15) -8*a(n-17) for n>18 %e A298707 All solutions for n=5 %e A298707 ..0..1..1. .0..0..1. .0..0..1. .0..1..0. .0..1..0. .0..1..1 %e A298707 ..0..1..1. .0..0..1. .0..0..1. .0..1..0. .0..1..0. .0..1..1 %e A298707 ..0..0..0. .0..1..1. .1..1..1. .1..1..1. .0..0..0. .0..0..1 %e A298707 ..0..1..1. .0..0..1. .0..0..1. .0..1..0. .0..1..0. .0..1..1 %e A298707 ..0..1..1. .0..0..1. .0..0..1. .0..1..0. .0..1..0. .0..1..1 %Y A298707 Cf. A298712. %K A298707 nonn,new %O A298707 1,2 %A A298707 _R. H. Hardin_, Jan 25 2018 %I A298706 %S A298706 0,4,0,11,781,3950,17377,8695221,359275256,26141521994 %N A298706 Number of nXn 0..1 arrays with every element equal to 1, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298706 Diagonal of A298712. %e A298706 Some solutions for n=5 %e A298706 ..0..1..1..0..0. .0..1..1..0..1. .0..1..0..0..1. .0..1..1..0..1 %e A298706 ..1..0..1..1..1. .0..1..1..0..1. .1..0..0..0..1. .0..1..1..0..1 %e A298706 ..1..1..1..1..1. .1..1..1..1..1. .0..0..0..0..0. .1..1..1..1..1 %e A298706 ..0..1..1..0..0. .0..1..1..1..0. .0..1..0..1..0. .0..1..1..1..1 %e A298706 ..0..1..1..1..1. .0..1..1..0..1. .0..1..0..1..0. .1..0..1..0..0 %Y A298706 Cf. A298712. %K A298706 nonn,new %O A298706 1,2 %A A298706 _R. H. Hardin_, Jan 25 2018 %I A298608 %S A298608 1,0,1,2,1,1,2,6,2,1,6,9,12,3,1,8,30,24,20,4,1,20,50,90,50,30,5,1,30, %T A298608 140,180,210,90,42,6,1,70,245,560,490,420,147,56,7,1,112,630,1120, %U A298608 1680,1120,756,224,72,8,1 %N A298608 Polynomials related to the Motzkin numbers for Coxeter type D, T(n, k) for n >= 0 and 0 <= k <= n. %C A298608 The polynomials evaluated at x = 1 give the analog of the Motzkin numbers for Coxeter type D (see A298300 (with a shift in the indexing)). %F A298608 T(n,k) = A109187(n,k) + A298609(n,k). %F A298608 The polynomials are defined by p(0, x) = 1 and for n >= 1 by p(n, x) = G(n,-n,-x/2) + G(n-1,-n,-x/2)*(n-1)/n where G(n, a, x) denotes the n-th Gegenbauer polynomial. %F A298608 p(n, x) = binomial(2*n,n)*(hypergeom([-n,-n], [-n+1/2], 1/2-x/4) + ((n-1)/(n+1))* hypergeom([-n+1,-n-1], [-n+1/2], 1/2-x/4))) for n >= 1. %e A298608 The first few polynomials are: %e A298608 p0(x) = 1; %e A298608 p1(x) = 0 + x; %e A298608 p2(x) = 2 + x + x^2; %e A298608 p3(x) = 2 + 6*x + 2*x^2 + x^3; %e A298608 p4(x) = 6 + 9*x + 12*x^2 + 3*x^3 + x^4; %e A298608 p5(x) = 8 + 30*x + 24*x^2 + 20*x^3 + 4*x^4 + x^5; %e A298608 p6(x) = 20 + 50*x + 90*x^2 + 50*x^3 + 30*x^4 + 5*x^5 + x^6; %e A298608 p7(x) = 30 + 140*x + 180*x^2 + 210*x^3 + 90*x^4 + 42*x^5 + 6*x^6 + x^7; %e A298608 The triangle starts: %e A298608 [0][ 1] %e A298608 [1][ 0, 1] %e A298608 [2][ 2, 1, 1] %e A298608 [3][ 2, 6, 2, 1] %e A298608 [4][ 6, 9, 12, 3, 1] %e A298608 [5][ 8, 30, 24, 20, 4, 1] %e A298608 [6][ 20, 50, 90, 50, 30, 5, 1] %e A298608 [7][ 30, 140, 180, 210, 90, 42, 6, 1] %e A298608 [8][ 70, 245, 560, 490, 420, 147, 56, 7, 1] %e A298608 [9][112, 630, 1120, 1680, 1120, 756, 224, 72, 8, 1] %p A298608 A298608Poly := n -> `if`(n=0, 1, binomial(2*n, n)*(hypergeom([-n, -n], [-n+1/2], 1/2-x/4) + ((n-1)/(n+1))*hypergeom([-n+1, -n-1], [-n+1/2], 1/2-x/4))): %p A298608 A298608Row := n -> op(PolynomialTools:-CoefficientList(simplify(A298608Poly(n)), x)): seq(A298608Row(n), n=0..9); %t A298608 p[0] := 1; %t A298608 p[n_] := GegenbauerC[n, -n , -x/2] + GegenbauerC[n - 1, -n , -x/2] (n - 1) / n; %t A298608 Table[CoefficientList[p[n], x], {n, 0, 9}] // Flatten %Y A298608 Row sums are A298300(n+1) for n >= 1. %Y A298608 Cf. A109187, A298609. %K A298608 nonn,tabl,new %O A298608 0,4 %A A298608 _Peter Luschny_, Jan 23 2018 %I A298609 %S A298609 0,0,0,0,1,0,2,0,2,0,0,9,0,3,0,8,0,24,0,4,0,0,50,0,50,0,5,0,30,0,180, %T A298609 0,90,0,6,0,0,245,0,490,0,147,0,7,0,112,0,1120,0,1120,0,224,0,8,0,0, %U A298609 1134,0,3780,0,2268,0,324,0,9,0,420,0,6300,0,10500,0,4200,0,450,0,10,0 %N A298609 Polynomials related to the Motzkin sums for Coxeter type D, T(n, k) for n >= 0 and 0 <= k <= n. %C A298609 The polynomials evaluated at x = 1 give the analog of the Motzkin sums for Coxeter type D (see A290380 (with a shift in the indexing)). %F A298609 A298608(n,k) = A109187(n,k) + T(n,k). %F A298609 The polynomials are defined by p(0, x) = p(1, x) = 0 and for n >= 2 by p(n, x) = G(n - 1, -n, -x/2)*(n - 1)/n where G(n, a, x) denotes the n-th Gegenbauer polynomial. %F A298609 p(n, x) = Catalan(n)*(n-1)*hypergeom([1-n, -n-1], [-n+1/2], 1/2-x/4) for n >= 2. %e A298609 The first few polynomials are: %e A298609 p0(x) = 0; %e A298609 p1(x) = 0; %e A298609 p2(x) = x; %e A298609 p3(x) = 2 + 2*x^2; %e A298609 p4(x) = 9*x + 3*x^3; %e A298609 p5(x) = 8 + 24*x^2 + 4*x^4; %e A298609 p6(x) = 50*x + 50*x^3 + 5*x^5; %e A298609 p7(x) = 30 + 180*x^2 + 90*x^4 + 6*x^6; %e A298609 p8(x) = 245*x + 490*x^3 + 147*x^5 + 7*x^7; %e A298609 p9(x) = 112 + 1120*x^2 + 1120*x^4 + 224*x^6 + 8*x^8; %e A298609 The triangle of coefficients extended by the main diagonal with zeros starts: %e A298609 [0][ 0] %e A298609 [1][ 0, 0] %e A298609 [2][ 0, 1, 0] %e A298609 [3][ 2, 0, 2, 0] %e A298609 [4][ 0, 9, 0, 3, 0] %e A298609 [5][ 8, 0, 24, 0, 4, 0] %e A298609 [6][ 0, 50, 0, 50, 0, 5, 0] %e A298609 [7][ 30, 0, 180, 0, 90, 0, 6, 0] %e A298609 [8][ 0, 245, 0, 490, 0, 147, 0, 7, 0] %e A298609 [9][112, 0, 1120, 0, 1120, 0, 224, 0, 8, 0] %p A298609 A298609Poly := n -> `if`(n<=1, 0, binomial(2*n, n)*((n-1)/(n+1))*hypergeom([1-n, -n-1], [-n+1/2], 1/2-x/4)): %p A298609 A298609Row := n -> if n=0 then 0 elif n=1 then 0,0 else op(PolynomialTools:-CoefficientList(simplify(A298609Poly(n)), x)),0 fi: %p A298609 seq(A298609Row(n), n=0..11); %t A298609 P298609[n_] := If[n <= 1, 0, GegenbauerC[n - 1, -n, -x/2] (n - 1)/n]; %t A298609 Flatten[ Join[ {{0}, {0, 0}}, %t A298609 Table[ Join[ CoefficientList[ P298609[n], x], {0}], {n, 2, 10}]]] %Y A298609 Cf. A109187, A290380, A298608. %K A298609 nonn,tabl,new %O A298609 0,7 %A A298609 _Peter Luschny_, Jan 23 2018 %I A298231 %S A298231 1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,2,2, %T A298231 1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,1,2, %U A298231 2,1,2,2,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,1,2,2,1,2,2,1,2,2,1,1,2,2,1,2,2,1,2,2,1 %N A298231 Fixed point of the morphism 1->1221, 2->122. %C A298231 This is the morphism in standard form. Its mirrored version 1->211, 2->2112 has fixed point 2,1,1,2,2,1,1,2,1,1,2,1,1,2,2,1,1,2,2,1,1,2,1,1,2,... which is the sequence of first differences of A284895. This can be seen by studying A284893, the fixed point x of the morphism sigma 0->01, 1->0111. Then x is also fixed point of sigma^2: 0->010111, 1->01011101110111. %C A298231 Note that x is a concatenation of 01 and 0111, and both words are always followed by 01, the common prefix of 01 and 0111. This implies that the two sigma^2 words induce increments in the positions of 1, which are respectively 2,1,1,2 and 2,1,1,2,1,1,2,1,1,2. This implies that 1->211, 2->2112 generates the sequence of first differences. %t A298231 Nest[Flatten[# /. {1 -> {1, 2, 2, 1}, 2 -> {1, 2, 2}}] &, {1}, 8] (* _Michael De Vlieger_, Jan 22 2018 *) %Y A298231 Cf. A284893, A284893, A284895. %K A298231 nonn,new %O A298231 1,2 %A A298231 _Michel Dekking_, Jan 15 2018 %I A298644 %S A298644 1,3,4,6,7,8,9,14,15,16,17,24,27,28,30,31,32,33,35,36,39,48,49,54,55, %T A298644 57,59,60,62,63,64,65,67,68,70,72,73,78,79,96,97,99,110,111,112,118, %U A298644 119,120,121,123,124,126,127,128,129,131,132,134,135,136,137,143,144,145,156,158 %N A298644 The indices of the Carlitz compositions (i.e., compositions without adjacent equal parts). %C A298644 We define the index of a composition to be the positive integer whose binary form has run-lengths (i.e., runs of 1's, runs of 0's, etc., from left to right) equal to the parts of the composition. Example: the composition [1,1,3,1] has index 46 since the binary form of 46 is 101110. %C A298644 The command c(n) from the Maple program yields the composition having index n. %e A298644 135 is in the sequence since its binary form is 10000111 and the composition [1,4,3] has no adjacent equal parts. %e A298644 139 is not in the sequence since its binary form is 10001011 and the composition [1,3,1,1,2] has two adjacent equal parts. %p A298644 Runs := proc (L) local j, r, i, k: j := 1: r[j] := L[1]: for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1: r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc: RunLengths := proc (L) map(nops, Runs(L)) end proc: c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc: pd := proc (n) options operator, arrow: product(c(n)[j]-c(n)[j+1], j = 1 .. nops(c(n))-1) end proc: A := {}; for n to 200 do if pd(n) <> 0 then A := `union`(A, {n}) else end if end do: A; # most of the Maple program is due to _W. Edwin Clark_ %t A298644 With[{nn = 18}, TakeWhile[#, # <= Floor[2^(2 + nn/Log2[nn])] &] &@ Union@ Apply[Join, #] &@ Table[Map[FromDigits[#, 2] &@ Flatten@ MapIndexed[ConstantArray[Boole@ OddQ@ #2, #1] &, #] &, Select[Map[Flatten[Map[# /. w_List :> If[First@ w == 1, Length@ w + 1, ConstantArray[1, Length@ w]] &, Split@ #] /. {a__, b_List, c__} :> {a, Most@ b, c}] &@ PadLeft[#, n - 1] &, IntegerDigits[Range[0, 2^n - 1], 2]], FreeQ[Differences@ #, 0] &]], {n, 2, nn}]] (* _Michael De Vlieger_, Jan 24 2018 *) %Y A298644 Cf. A003242, A101211. %K A298644 nonn,new %O A298644 1,2 %A A298644 _Emeric Deutsch_, Jan 24 2018 %I A298691 %S A298691 1,1,3,17,144,1647,24037,429483,9088749,221942779,6130801041, %T A298691 188708846991,6398116247554,236786117903526,9495515095867953, %U A298691 410104221125229354,18977504682428845671,936731766873748776822,49127713187418767376060,2728178479576867266738579,159924801506251429348644138,9868564065320443974954599471 %N A298691 G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n*(n+1)/2, n) * x^n / A(x)^( n*(n-1)/2 ). %e A298691 G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 144*x^4 + 1647*x^5 + 24037*x^6 + 429483*x^7 + 9088749*x^8 + 221942779*x^9 + 6130801041*x^10 + 188708846991*x^11 + 6398116247554*x^12 + 236786117903526*x^13 + 9495515095867953*x^14 + 410104221125229354*x^15 + ... %e A298691 such that %e A298691 A(x) = 1 + C(1,1)*x + C(3,2)*x^2/A(x) + C(6,3)*x^3/A(x)^3 + C(10,4)*x^4/A(x)^6 + C(15,5)*x^5/A(x)^10 + C(21,6)*x^6/A(x)^15 + C(28,7)*x^7/A(x)^21 + ... %e A298691 more explicitly, %e A298691 A(x) = 1 + x + 3*x^2/A(x) + 20*x^3/A(x)^3 + 210*x^4/A(x)^6 + 3003*x^5/A(x)^10 + 54264*x^6/A(x)^15 + 1184040*x^7/A(x)^21 + 30260340*x^8/A(x)^28 + ... %o A298691 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = Vec(sum(m=0,#A,binomial(m*(m+1)/2,m) * x^m/Ser(A)^(m*(m-1)/2) ))); A[n+1]} %o A298691 for(n=0,30,print1(a(n),", ")) %Y A298691 Cf. A014068, A298689, A298690. %K A298691 nonn,new %O A298691 0,3 %A A298691 _Paul D. Hanna_, Jan 24 2018 %I A298690 %S A298690 1,1,2,10,83,971,14679,271065,5887674,146573343,4106195739, %T A298690 127709962780,4364136955874,162503129082497,6548680061635319, %U A298690 283973223632787150,13185195626147207058,652695122347799336199,34316223642036784123819,1909798106976656110119169,112165977515060359849066878,6933265352057611483132200642 %N A298690 G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n*(n+1)/2, n) * x^n / A(x)^( n*(n+1)/2 ). %e A298690 G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 83*x^4 + 971*x^5 + 14679*x^6 + 271065*x^7 + 5887674*x^8 + 146573343*x^9 + 4106195739*x^10 + 127709962780*x^11 + 4364136955874*x^12 + 162503129082497*x^13 + 6548680061635319*x^14 + 283973223632787150*x^15 + ... %e A298690 such that %e A298690 A(x) = 1 + C(1,1)*x/A(x) + C(3,2)*x^2/A(x)^3 + C(6,3)*x^3/A(x)^6 + C(10,4)*x^4/A(x)^10 + C(15,5)*x^5/A(x)^15 + C(21,6)*x^6/A(x)^21 + C(28,7)*x^7/A(x)^28 + ... %e A298690 more explicitly, %e A298690 A(x) = 1 + x/A(x) + 3*x^2/A(x)^3 + 20*x^3/A(x)^6 + 210*x^4/A(x)^10 + 3003*x^5/A(x)^15 + 54264*x^6/A(x)^21 + 1184040*x^7/A(x)^28 + 30260340*x^8/A(x)^36 + ... %o A298690 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = Vec(sum(m=0,#A,binomial(m*(m+1)/2,m) * x^m/Ser(A)^(m*(m+1)/2) ))); G=Ser(A); A[n+1]} %o A298690 for(n=0,30,print1(a(n),", ")) %Y A298690 Cf. A014068, A298689, A298691. %K A298690 nonn,new %O A298690 0,3 %A A298690 _Paul D. Hanna_, Jan 24 2018 %I A298689 %S A298689 1,1,5,56,957,22312,666666,24367474,1051351629,52144520972, %T A298689 2915915251326,181227240764128,12382862552065170,922234506009645794, %U A298689 74345308066436693828,6449466281781165675666,599083515375854753327365,59328642583049975996828036,6240245388930730524658068558,694754212357547941002786433000,81628078642468462576697539116234 %N A298689 G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n^2, n) * x^n / A(x)^( n^2 ). %C A298689 Compare to: Sum_{n>=0} C(m*n,n) * x^n / (1+x)^(m*n) = (1+x)/(1 - (m-1)*x) holds for fixed m. %H A298689 Paul D. Hanna, Table of n, a(n) for n = 0..260 %F A298689 a(2^k) is odd for k>=0, and a(n) is even elsewhere except at n=0 (conjecture). %e A298689 G.f.: A(x) = 1 + x + 5*x^2 + 56*x^3 + 957*x^4 + 22312*x^5 + 666666*x^6 + 24367474*x^7 + 1051351629*x^8 + 52144520972*x^9 + 2915915251326*x^10 + 181227240764128*x^11 + 12382862552065170*x^12 + ... %e A298689 such that %e A298689 A(x) = 1 + C(1,1)*x/A(x) + C(4,2)*x^2/A(x)^4 + C(9,3)*x^3/A(x)^9 + C(16,4)*x^4/A(x)^16 + C(25,5)*x^5/A(x)^25 + C(36,6)*x^6/A(x)^36 + C(49,7)*x^7/A(x)^49 + ... %e A298689 more explicitly, %e A298689 A(x) = 1 + x/A(x) + 6*x^2/A(x)^4 + 84*x^3/A(x)^9 + 1820*x^4/A(x)^16 + 53130*x^5/A(x)^25 + 1947792*x^6/A(x)^36 + 85900584*x^7/A(x)^49 + ... %o A298689 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = Vec(sum(m=0,#A,binomial(m^2,m) * x^m/Ser(A)^(m^2) ))); A[n+1]} %o A298689 for(n=0,30,print1(a(n),", ")) %Y A298689 Cf. A014062, A298690, A298691. %K A298689 nonn,new %O A298689 0,3 %A A298689 _Paul D. Hanna_, Jan 24 2018 %I A296231 %S A296231 1,1,1,2,3,7,16,48,157,586,2362,10214,46672,223752,1118799,5810185, %T A296231 31237145,173412537,992006284,5837461604,35283954583,218791917313, %U A296231 1390314155401,9044905749879,60190822583318,409404760891303,2844213921090065,20168470493811065,145888129690256442,1075859539461621404,8084389249391405645,61869341164985700882,481984158600673224200 %N A296231 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n / (1-x)^( n*(n+1) ) / A(x)^( (n+1)*(n+2)/2 ). %H A296231 Paul D. Hanna, Table of n, a(n) for n = 0..200 %e A296231 G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 7*x^5 + 16*x^6 + 48*x^7 + 157*x^8 + 586*x^9 + 2362*x^10 + 10214*x^11 + 46672*x^12 + 223752*x^13+ 1118799*x^14 + 5810185*x^15 + ... %e A296231 such that %e A296231 1 = 1/A(x) + x/(1-x)^2/A(x)^3 + x^2/(1-x)^6/A(x)^6 + x^3/(1-x)^12/A(x)^10 + x^4/(1-x)^20/A(x)^15 + x^5/(1-x)^30/A(x)^21 + x^6/(1-x)^42/A(x)^28 + x^7/(1-x)^56/A(x)^36 + ... %o A296231 (PARI) {a(n) = my(A=[1],V); for(i=0,n, A = concat(A,0); V = Vec(sum(n=0,#A,1/(1-x +x*O(x^#A))^(n*(n+1))*x^n/Ser(A)^((n+1)*(n+2)/2)) ); A[#A]=V[#A] ); A[n+1]} %o A296231 for(n=0,30,print1(a(n),", ")) %Y A296231 Cf. A296230. %K A296231 nonn,new %O A296231 0,4 %A A296231 _Paul D. Hanna_, Jan 24 2018 %I A296230 %S A296230 1,1,0,1,0,1,1,0,4,0,6,13,9,48,101,147,542,1244,2385,8158,19191,44960, %T A296230 145355,356921,953648,2971797,7728368,22395844,68642687,189610373, %U A296230 577526057,1770461983,5170947386,16264118299,50488278032,154687144811,498055705248,1577949582705,5029555661992,16520308729413,53633742931559,176588771399224 %N A296230 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n / (1-x)^( n*(n+1)/2 ) / A(x)^( (n+1)*(n+2)/2 ). %H A296230 Paul D. Hanna, Table of n, a(n) for n = 0..200 %e A296230 G.f.: A(x) = 1 + x + x^3 + x^5 + x^6 + 4*x^8 + 6*x^10 + 13*x^11 + 9*x^12 + 48*x^13 + 101*x^14 + 147*x^15 + 542*x^16 + 1244*x^17 + 2385*x^18 + 8158*x^19 + 19191*x^20 + ... %e A296230 such that %e A296230 1 = 1/A(x) + x/(1-x)/A(x)^3 + x^2/(1-x)^3/A(x)^6 + x^3/(1-x)^6/A(x)^10 + x^4/(1-x)^10/A(x)^15 + x^5/(1-x)^15/A(x)^21 + x^6/(1-x)^21/A(x)^28 + x^7/(1-x)^28/A(x)^36 + ... %e A296230 Compare to the trivial identity: %e A296230 1 = 1/(1+x) + x*(1+x)/(1+x)^3 + x^2*(1+x)^3/(1+x)^6 + x^3*(1+x)^6/(1+x)^10 + x^4*(1+x)^10/(1+x)^15 + x^5*(1+x)^15/(1+x)^21 + ... %o A296230 (PARI) {a(n) = my(A=[1],V); for(i=0,n, A = concat(A,0); V = Vec(sum(n=0,#A,1/(1-x +x*O(x^#A))^(n*(n+1)/2)*x^n/Ser(A)^((n+1)*(n+2)/2)) ); A[#A]=V[#A] ); A[n+1]} %o A296230 for(n=0,40,print1(a(n),", ")) %Y A296230 Cf. A296231. %K A296230 nonn,new %O A296230 0,9 %A A296230 _Paul D. Hanna_, Jan 24 2018 %I A298668 %S A298668 1,0,1,0,1,0,1,1,0,1,3,0,1,7,2,0,1,15,12,0,1,31,50,6,0,1,63,180,60,0, %T A298668 1,127,602,390,24,0,1,255,1932,2100,360,0,1,511,6050,10206,3360,120,0, %U A298668 1,1023,18660,46620,25200,2520,0,1,2047,57002,204630,166824,31920,720 %N A298668 Number T(n,k) of set partitions of [n] into k blocks such that the absolute difference between least elements of consecutive blocks is always > 1; triangle T(n,k), n>=0, 0<=k<=ceiling(n/2), read by rows. %H A298668 Alois P. Heinz, Rows n = 0..200, flattened %F A298668 T(n,k) = (k-1)! * Stirling2(n-k+1,k) for k>0, T(n,0) = A000007(n). %F A298668 T(n,k) = Sum_{j=0..k-1} (-1)^j*C(k-1,j)*(k-j)^(n-k) for k>0, T(n,0) = A000007(n). %F A298668 Sum_{j>=0} T(n+j,j) = A076726(n) = 2*A000670(n) = A000629(n) + A000007(n). %e A298668 T(5,1) = 1: 12345. %e A298668 T(5,2) = 7: 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345. %e A298668 T(5,3) = 2: 124|3|5, 12|34|5. %e A298668 T(7,4) = 6: 1246|3|5|7, 124|36|5|7, 124|3|56|7, 126|34|5|7, 12|346|5|7, 12|34|56|7. %e A298668 T(9,5) = 24: 12468|3|5|7|9, 1246|38|5|7|9, 1246|3|58|7|9, 1246|3|5|78|9, 1248|36|5|7|9, 124|368|5|7|9, 124|36|58|7|9, 124|36|5|78|9, 1248|3|56|7|9, 124|38|56|7|9, 124|3|568|7|9, 124|3|56|78|9, 1268|34|5|7|9, 126|348|5|7|9, 126|34|58|7|9, 126|34|5|78|9, 128|346|5|7|9, 12|3468|5|7|9, 12|346|58|7|9, 12|346|5|78|9, 128|34|56|7|9, 12|348|56|7|9, 12|34|568|7|9, 12|34|56|78|9. %e A298668 Triangle T(n,k) begins: %e A298668 1; %e A298668 0, 1; %e A298668 0, 1; %e A298668 0, 1, 1; %e A298668 0, 1, 3; %e A298668 0, 1, 7, 2; %e A298668 0, 1, 15, 12; %e A298668 0, 1, 31, 50, 6; %e A298668 0, 1, 63, 180, 60; %e A298668 0, 1, 127, 602, 390, 24; %e A298668 0, 1, 255, 1932, 2100, 360; %e A298668 0, 1, 511, 6050, 10206, 3360, 120; %e A298668 0, 1, 1023, 18660, 46620, 25200, 2520; %p A298668 b:= proc(n, m, t) option remember; `if`(n=0, x^m, add( %p A298668 b(n-1, max(m, j), `if`(j>m, 1, 0)), j=1..m+1-t)) %p A298668 end: %p A298668 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)): %p A298668 seq(T(n), n=0..14); %p A298668 # second Maple program: %p A298668 T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), (k-1)!*Stirling2(n-k+1, k)): %p A298668 seq(seq(T(n, k), k=0..ceil(n/2)), n=0..14); %p A298668 # third Maple program: %p A298668 T:= proc(n, k) option remember; `if`(k<2, `if`(n=0 xor k=0, 0, 1), %p A298668 `if`(k>ceil(n/2), 0, add((k-j)*T(n-1-j, k-j), j=0..1))) %p A298668 end: %p A298668 seq(seq(T(n, k), k=0..ceil(n/2)), n=0..14); %Y A298668 Columns k=0-11 give (offsets may differ): A000007, A057427, A168604, A028243, A028244, A028245, A032180, A228909, A228910, A228911, A228912, A228913. %Y A298668 Row sums give A229046(n-1) for n>0. %Y A298668 T(2n+1,n+1) gives A000142. %Y A298668 T(2n,n) gives A001710(n+1). %Y A298668 Cf. A000629, A000670, A008277, A028246, A048993, A076726, A110654, A173018. %K A298668 nonn,tabf,new %O A298668 0,11 %A A298668 _Alois P. Heinz_, Jan 24 2018 %I A296012 %S A296012 79,101,103,149,151,167,191,193,227,229,257,277,281,283,347,349,353, %T A296012 359,367,373,401,431,433,439,461,463,479,509,557,563,607,613,617,619, %U A296012 641,643,647,653,659,661,709,733,739,743,761,797,821,823,857,859,863,887,907,911,967,971,977,983,1019,1021 %N A296012 Primes of the form k + k+1 + k+2 +-1 where k, k+1, and k+2 are all composite numbers. %C A296012 Primes p such that floor((p-2)/3) and floor((p-2)/3)+2 are composite. - _Robert Israel_, Dec 03 2017 %H A296012 Chai Wah Wu, Table of n, a(n) for n = 1..10000 %e A296012 25 + 26 + 27 + 1 = 79, %e A296012 33 + 34 + 35 - 1 = 101, %e A296012 33 + 34 + 35 + 1 = 103, etc. %p A296012 filter:= proc(n) local k; %p A296012 if not isprime(n) then return false fi; %p A296012 k:= floor((n-2)/3); %p A296012 not isprime(k) and not isprime(k+1) and not isprime(k+2) %p A296012 end proc: %p A296012 select(filter, [seq(i,i=5..2000, 2)]); # _Robert Israel_, Dec 03 2017 %t A296012 Select[Join @@ Map[{{Total@ # - 1, #}, {Total@ # + 1, #}} &, Partition[Range@ 350, 3, 1]], And[PrimeQ@ First@ #, AllTrue[Last@ #, CompositeQ]] &][[All, 1]] (* _Michael De Vlieger_, Dec 03 2017 *) %o A296012 (Python) %o A296012 from __future__ import division %o A296012 from sympy import nextprime, isprime %o A296012 A296012_list, p = [], 2 %o A296012 while len(A296012_list) < 10000: %o A296012 k = (p-2)//3 %o A296012 if not (isprime(k) or isprime(k+2)): %o A296012 A296012_list.append(p) %o A296012 p = nextprime(p) # _Chai Wah Wu_, Jan 24 2018 %Y A296012 Cf. A000045, A136799. %K A296012 nonn,new %O A296012 1,1 %A A296012 _Martin Michael Musatov_, Dec 02 2017 %I A298667 %S A298667 1,1,1,1,1,1,1,0,0,1,1,1,3,1,1,1,2,1,1,2,1,1,1,5,2,5,1,1,1,8,8,2,2,8, %T A298667 8,1,1,5,7,9,20,9,7,5,1,1,22,25,13,38,38,13,25,22,1,1,29,25,26,84,169, %U A298667 84,26,25,29,1,1,60,58,74,319,238,238,319,74,58,60,1,1,121,95,134,770,1019,978 %N A298667 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298667 Table starts %C A298667 .1..1..1..1...1....1.....1.....1......1.......1.......1........1.........1 %C A298667 .1..1..0..1...2....1.....8.....5.....22......29......60......121.......194 %C A298667 .1..0..3..1...5....8.....7....25.....25......58......95......155.......299 %C A298667 .1..1..1..2...2....9....13....26.....74.....134.....325......731......1568 %C A298667 .1..2..5..2..20...38....84...319....770....2106....6301....17282.....47774 %C A298667 .1..1..8..9..38..169...238..1019...3112....7518...25887....74736....218787 %C A298667 .1..8..7.13..84..238...978..3895..11843...49752..184380...640323...2550161 %C A298667 .1..5.25.26.319.1019..3895.22938..81474..386923.1791871..7595552..35648606 %C A298667 .1.22.25.74.770.3112.11843.81474.329249.1693779.9364144.43238329.235861179 %H A298667 R. H. Hardin, Table of n, a(n) for n = 1..241 %F A298667 Empirical for column k: %F A298667 k=1: a(n) = a(n-1) %F A298667 k=2: a(n) = 3*a(n-2) +2*a(n-3) -2*a(n-4) for n>5 %F A298667 k=3: [order 12] %F A298667 k=4: [order 35] %e A298667 Some solutions for n=8 k=4 %e A298667 ..0..1..1..0. .0..1..1..0. .0..1..1..0. .0..1..1..0. .0..1..1..0 %e A298667 ..1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1 %e A298667 ..0..1..1..1. .0..1..1..0. .0..1..1..0. .0..1..1..0. .0..1..1..1 %e A298667 ..1..1..1..0. .1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..0..1 %e A298667 ..1..1..1..1. .1..1..1..1. .1..0..1..1. .1..1..0..1. .1..1..1..1 %e A298667 ..0..1..1..0. .0..1..1..0. .1..1..1..0. .0..1..1..1. .0..1..1..0 %e A298667 ..1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1 %e A298667 ..0..1..1..0. .0..1..1..0. .0..1..1..0. .0..1..1..0. .0..1..1..0 %Y A298667 Column 2 is A297809(n-2). %K A298667 nonn,tabl,new %O A298667 1,13 %A A298667 _R. H. Hardin_, Jan 24 2018 %I A298666 %S A298666 1,8,7,13,84,238,978,3895,11843,49752,184380,640323,2550161,9291605, %T A298666 34411340,132207077,488538547,1844610495,6998077491,26247637676, %U A298666 99671069188,378042682211,1432299046200,5458358304491,20789207845451 %N A298666 Number of nX7 0..1 arrays with every element equal to 0, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298666 Column 7 of A298667. %H A298666 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298666 Some solutions for n=8 %e A298666 ..0..1..1..1..1..1..0. .0..1..0..1..0..1..0. .0..0..1..0..0..0..0 %e A298666 ..1..1..1..0..0..1..1. .1..1..1..1..1..1..1. .0..0..0..0..1..0..0 %e A298666 ..0..1..1..0..0..0..1. .1..1..1..1..1..1..1. .0..1..0..0..0..0..1 %e A298666 ..1..1..0..0..0..0..1. .0..1..1..0..1..0..1. .0..0..0..1..0..0..0 %e A298666 ..1..1..1..0..0..1..1. .1..1..1..1..1..1..1. .0..0..0..0..0..1..0 %e A298666 ..1..0..1..1..1..1..0. .0..1..1..1..1..1..0. .1..0..0..0..0..0..0 %e A298666 ..1..1..1..1..1..1..1. .1..1..1..0..1..1..1. .0..0..1..0..0..0..0 %e A298666 ..1..1..0..1..0..1..1. .0..1..1..1..1..1..0. .0..0..0..0..1..0..1 %Y A298666 Cf. A298667. %K A298666 nonn,new %O A298666 1,2 %A A298666 _R. H. Hardin_, Jan 24 2018 %I A298665 %S A298665 1,1,8,9,38,169,238,1019,3112,7518,25887,74736,218787,690826,2069920, %T A298665 6363967,19855161,61528839,192963096,609853755,1933979526,6175400889, %U A298665 19859400627,64164744962,208511225451,681665542190,2238543624216 %N A298665 Number of nX6 0..1 arrays with every element equal to 0, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298665 Column 6 of A298667. %H A298665 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298665 Some solutions for n=8 %e A298665 ..0..1..1..0..1..0. .0..0..1..0..0..1. .0..0..0..0..0..1. .0..0..1..1..0..0 %e A298665 ..1..1..1..1..1..1. .0..0..0..0..0..0. .0..0..1..0..0..0. .0..0..1..1..0..0 %e A298665 ..1..0..1..1..1..1. .0..1..0..0..1..0. .1..0..0..0..0..0. .0..1..1..1..1..0 %e A298665 ..1..1..1..1..0..1. .0..0..0..0..0..0. .0..0..0..0..0..1. .0..1..1..1..0..0 %e A298665 ..1..0..1..1..1..1. .1..0..0..1..0..0. .0..0..1..0..0..0. .0..0..1..1..0..0 %e A298665 ..1..1..1..1..1..0. .0..0..0..0..0..1. .1..0..0..0..1..0. .0..1..1..1..1..0 %e A298665 ..1..1..1..0..1..1. .0..0..0..0..0..0. .0..0..0..0..0..0. .0..0..1..1..0..0 %e A298665 ..0..1..1..1..1..1. .1..0..0..1..0..0. .1..0..0..1..0..0. .0..0..1..1..0..0 %Y A298665 Cf. A298667. %K A298665 nonn,new %O A298665 1,3 %A A298665 _R. H. Hardin_, Jan 24 2018 %I A298664 %S A298664 1,2,5,2,20,38,84,319,770,2106,6301,17282,47774,137186,386924,1083606, %T A298664 3099891,8800002,24988837,71554943,204638340,586541392,1687246083, %U A298664 4861247458,14034796678,40624856818,117836193317,342470517019 %N A298664 Number of nX5 0..1 arrays with every element equal to 0, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298664 Column 5 of A298667. %H A298664 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298664 Some solutions for n=8 %e A298664 ..0..1..1..1..0. .0..0..0..0..0. .0..1..0..1..0. .0..1..1..1..0 %e A298664 ..1..1..0..1..1. .0..0..1..0..0. .1..1..1..1..1. .1..1..0..1..1 %e A298664 ..1..0..0..0..1. .1..1..1..1..1. .1..1..1..1..1. .1..1..1..1..1 %e A298664 ..1..1..0..1..1. .1..1..1..1..1. .0..1..1..0..1. .0..1..1..0..1 %e A298664 ..1..0..0..0..1. .0..1..1..0..1. .1..1..1..1..1. .1..1..1..1..1 %e A298664 ..1..0..0..0..1. .1..1..1..1..1. .0..1..1..1..0. .1..0..1..1..0 %e A298664 ..1..1..0..1..1. .1..1..1..1..1. .1..1..0..1..1. .1..1..1..1..1 %e A298664 ..0..1..1..1..0. .0..1..0..1..0. .1..1..1..1..1. .1..1..0..1..1 %Y A298664 Cf. A298667. %K A298664 nonn,new %O A298664 1,2 %A A298664 _R. H. Hardin_, Jan 24 2018 %I A298663 %S A298663 1,1,1,2,2,9,13,26,74,134,325,731,1568,3625,8039,17982,40534,90659, %T A298663 203629,456958,1025608,2302082,5167454,11602659,26043008,58476012, %U A298663 131288529,294749848,661837914,1485958282,3336440686,7491565617 %N A298663 Number of nX4 0..1 arrays with every element equal to 0, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298663 Column 4 of A298667. %H A298663 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298663 Empirical: a(n) = 2*a(n-1) +3*a(n-2) +3*a(n-3) -19*a(n-4) -10*a(n-5) -3*a(n-6) +49*a(n-7) +21*a(n-8) +26*a(n-9) -57*a(n-10) -45*a(n-11) -44*a(n-12) +15*a(n-13) +27*a(n-14) +41*a(n-15) -4*a(n-16) +13*a(n-17) +32*a(n-18) -27*a(n-19) -87*a(n-20) -5*a(n-21) +52*a(n-22) +48*a(n-23) +24*a(n-24) +18*a(n-25) -27*a(n-26) -46*a(n-27) +11*a(n-28) +56*a(n-29) +5*a(n-30) -34*a(n-31) -21*a(n-32) -3*a(n-33) -a(n-34) +6*a(n-35) %e A298663 Some solutions for n=8 %e A298663 ..0..1..1..0. .0..1..1..0. .0..1..1..0. .0..1..1..0. .0..1..1..0 %e A298663 ..1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1 %e A298663 ..1..1..1..0. .0..1..1..1. .0..1..1..0. .0..1..1..0. .0..1..1..0 %e A298663 ..0..1..1..1. .1..1..1..0. .1..1..1..1. .1..1..1..1. .1..1..1..1 %e A298663 ..1..1..1..0. .0..1..1..1. .1..1..0..1. .0..1..1..0. .0..1..1..1 %e A298663 ..1..0..1..1. .1..1..1..0. .0..1..1..1. .1..1..1..1. .1..1..1..0 %e A298663 ..1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1. .1..1..1..1 %e A298663 ..0..1..1..0. .0..1..1..0. .0..1..1..0. .0..1..1..0. .0..1..1..0 %Y A298663 Cf. A298667. %K A298663 nonn,new %O A298663 1,4 %A A298663 _R. H. Hardin_, Jan 24 2018 %I A298662 %S A298662 1,0,3,1,5,8,7,25,25,58,95,155,299,494,905,1623,2867,5260,9421,17149, %T A298662 31225,56740,103795,189417,346621,634790,1162643,2132211,3910471, %U A298662 7175862,13172677,24185043,44417767,81586900,149883961,275388205,506024365 %N A298662 Number of nX3 0..1 arrays with every element equal to 0, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298662 Column 3 of A298667. %H A298662 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298662 Empirical: a(n) = a(n-1) +3*a(n-2) +a(n-3) -4*a(n-4) -6*a(n-5) -3*a(n-6) +5*a(n-7) +7*a(n-8) -a(n-9) +2*a(n-11) -2*a(n-12) %e A298662 Some solutions for n=8 %e A298662 ..0..1..0. .0..1..1. .0..1..0. .0..1..0. .0..1..1. .0..0..1. .0..1..0 %e A298662 ..1..1..1. .1..1..1. .1..1..1. .1..1..1. .1..1..1. .0..0..0. .1..1..1 %e A298662 ..1..1..1. .1..0..1. .0..1..0. .0..1..0. .1..0..1. .0..1..0. .1..1..1 %e A298662 ..1..0..0. .1..1..1. .1..1..1. .1..1..1. .1..1..1. .0..0..0. .0..0..1 %e A298662 ..1..0..0. .0..1..1. .0..1..0. .1..1..1. .1..1..0. .0..0..0. .0..0..1 %e A298662 ..1..1..1. .1..1..0. .1..1..1. .1..0..1. .0..1..1. .0..1..0. .1..1..1 %e A298662 ..1..1..1. .1..1..1. .1..1..1. .1..1..1. .1..1..1. .0..0..0. .1..1..1 %e A298662 ..0..1..0. .0..1..1. .0..1..0. .1..1..0. .1..1..0. .1..0..0. .0..1..0 %Y A298662 Cf. A298667. %K A298662 nonn,new %O A298662 1,3 %A A298662 _R. H. Hardin_, Jan 24 2018 %I A298661 %S A298661 1,1,3,2,20,169,978,22938,329249,12541065,628713833 %N A298661 Number of nXn 0..1 arrays with every element equal to 0, 3, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298661 Diagonal of A298667. %e A298661 Some solutions for n=8 %e A298661 ..0..0..1..0..0..0..0..1. .0..0..1..0..0..0..0..1. .0..0..0..0..1..0..0..1 %e A298661 ..0..0..0..0..1..0..0..0. .0..0..0..0..1..0..0..0. .0..0..1..0..0..0..0..0 %e A298661 ..0..1..0..0..0..0..0..1. .0..1..0..0..0..0..0..0. .1..0..0..0..0..0..0..0 %e A298661 ..0..0..0..0..1..0..0..0. .0..0..0..0..0..0..1..1. .0..0..0..0..1..0..1..1 %e A298661 ..0..1..0..0..0..0..1..0. .0..0..0..1..0..0..1..1. .0..0..1..1..1..1..1..1 %e A298661 ..0..0..0..0..0..0..0..0. .0..1..0..0..0..0..0..0. .1..1..1..1..1..1..1..0 %e A298661 ..0..0..1..0..0..1..0..0. .0..0..0..0..1..0..0..0. .1..1..1..1..0..1..1..1 %e A298661 ..1..0..0..0..0..0..0..1. .0..0..1..0..0..0..0..1. .0..1..0..1..1..1..1..0 %Y A298661 Cf. A298667. %K A298661 nonn,new %O A298661 1,3 %A A298661 _R. H. Hardin_, Jan 24 2018 %I A298637 %S A298637 1,2,3,1,4,4,5,9,2,6,16,10,7,25,27,5,8,36,56,28,9,49,100,84,14,10,64, %T A298637 162,192,84,11,81,245,375,270,42,12,100,352,660,660,264,13,121,486, %U A298637 1078,1375,891,132,14,144,650,1664,2574,2288,858,15,169,847,2457,4459,5005,3003,429 %N A298637 Triangular array of a Catalan number variety: T(n,k) is the number of words consisting of n parentheses containing k well-balanced pairs. %C A298637 A well-balanced run in a word of parentheses is a maximal run where every initial segment of the run has at least as many left parentheses as right ones and the number of open parentheses is the same as that of closed ones. The variable k in the sequence definition is the sum of the count of balanced pairs in all maximal runs in the word and n is the length of the word. Runs are maximal substrings counted by ordinary Catalan numbers. %H A298637 Alois P. Heinz, Rows n = 0..200, flattened %H A298637 Marko Riedel et al., Generalisation for Catalan number. %H A298637 Marko Riedel, Maple code for A298637 including enumeration, generating function, and two closed forms. %F A298637 T(n,k) = ((n+1-2*k)^2/(n+1))*C(n+1,k) where 0 <= k <= floor(n/2). %F A298637 Bivariate o.g.f. is C(u*z^2)/(1-z*C(u*z^2))^2 with u counting pairs of parentheses and z counting total word length where C(z) = (1-sqrt(1-4*z))/(2*z) is the o.g.f. of the Catalan numbers. %F A298637 T(2*k,k) = C(k), the k-th Catalan number. %F A298637 T(n,0) = n+1 by construction. %e A298637 The word ))))(()(()))((() contains five well-balanced pairs of parentheses. %e A298637 Triangular array T(n,k) begins: %e A298637 1; %e A298637 2; %e A298637 3, 1; %e A298637 4, 4; %e A298637 5, 9, 2; %e A298637 6, 16, 10; %e A298637 7, 25, 27, 5; %e A298637 8, 36, 56, 28; %e A298637 9, 49, 100, 84, 14; %e A298637 10, 64, 162, 192, 84; %e A298637 11, 81, 245, 375, 270, 42; %e A298637 12, 100, 352, 660, 660, 264; %p A298637 b:= proc(n, i) option remember; expand(`if`(n=0, 1, %p A298637 `if`(i>0, x, 1)*b(n-1, max(0, i-1))+b(n-1, i+1))) %p A298637 end: %p A298637 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)): %p A298637 seq(T(n), n=0..16); # _Alois P. Heinz_, Jan 23 2018 %t A298637 Table[((n + 1 - 2 k)^2/(n + 1)) Binomial[n + 1, k], {n, 0, 17}, {k, 0, Floor[n/2]}] // Flatten (* _Michael De Vlieger_, Jan 23 2018 *) %Y A298637 Row sums give A000079. %Y A298637 T(2n,n) gives A000108. %Y A298637 T(2n+1,n) gives A068875. T(n,1) gives A000290. T(2n,2) gives A280089. %Y A298637 Cf. A007318, A061554. %K A298637 nonn,tabf,new %O A298637 0,2 %A A298637 _Marko Riedel_, Jan 23 2018 %I A298660 %S A298660 0,1,1,1,3,1,2,7,7,2,3,13,15,13,3,5,23,19,19,23,5,8,49,23,40,23,49,8, %T A298660 13,95,34,85,85,34,95,13,21,177,63,173,177,173,63,177,21,34,359,96, %U A298660 322,431,431,322,96,359,34,55,705,147,635,876,1116,876,635,147,705,55,89,1351 %N A298660 T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298660 Table starts %C A298660 ..0...1...1....2....3.....5.....8.....13......21......34.......55........89 %C A298660 ..1...3...7...13...23....49....95....177.....359.....705.....1351......2689 %C A298660 ..1...7..15...19...23....34....63.....96.....147.....233......368.......588 %C A298660 ..2..13..19...40...85...173...322....635....1325....2806.....5877.....12293 %C A298660 ..3..23..23...85..177...431...876...2137....5002...11687....27591.....64253 %C A298660 ..5..49..34..173..431..1116..2562...6711...17405...48462...125671....334571 %C A298660 ..8..95..63..322..876..2562..7964..24801...74358..242072...745571...2349275 %C A298660 .13.177..96..635.2137..6711.24801..89543..322065.1213296..4468276..16453935 %C A298660 .21.359.147.1325.5002.17405.74358.322065.1367704.6098314.26543249.116098205 %H A298660 R. H. Hardin, Table of n, a(n) for n = 1..241 %F A298660 Empirical for column k: %F A298660 k=1: a(n) = a(n-1) +a(n-2) %F A298660 k=2: a(n) = 3*a(n-1) -2*a(n-2) +4*a(n-3) -10*a(n-4) +4*a(n-5) for n>6 %F A298660 k=3: [order 18] for n>19 %F A298660 k=4: [order 72] for n>73 %e A298660 Some solutions for n=5 k=4 %e A298660 ..0..1..1..0. .0..1..1..0. .0..0..1..0. .0..0..1..1. .0..1..0..0 %e A298660 ..0..0..0..0. .1..0..0..0. .1..0..1..0. .1..0..1..0. .0..1..0..1 %e A298660 ..0..0..0..0. .1..0..0..0. .1..1..1..1. .1..0..0..0. .1..1..1..1 %e A298660 ..0..1..0..1. .1..0..1..0. .1..1..1..1. .1..0..0..0. .1..1..1..1 %e A298660 ..1..1..0..1. .0..0..1..1. .1..0..0..1. .0..1..1..0. .1..0..0..1 %Y A298660 Column 1 is A000045(n-1). %Y A298660 Column 2 is A297852. %Y A298660 Column 3 is A298050. %K A298660 nonn,tabl,new %O A298660 1,5 %A A298660 _R. H. Hardin_, Jan 24 2018 %I A298659 %S A298659 8,95,63,322,876,2562,7964,24801,74358,242072,745571,2349275,7433849, %T A298659 23406485,73834463,233339001,735993576,2324359169,7339529089, %U A298659 23174991985,73178503209,231090268648,729735594047,2304427312579,7277098813048 %N A298659 Number of nX7 0..1 arrays with every element equal to 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298659 Column 7 of A298660. %H A298659 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298659 Some solutions for n=5 %e A298659 ..0..1..1..0..0..0..1. .0..0..1..0..0..0..1. .0..1..0..1..0..0..1 %e A298659 ..1..0..0..0..0..1..1. .1..1..0..0..0..1..1. .1..0..1..0..1..1..1 %e A298659 ..1..0..0..0..0..0..0. .0..0..0..0..0..0..0. .1..0..0..0..1..1..1 %e A298659 ..1..0..1..0..1..0..1. .1..0..1..0..1..0..1. .1..0..0..0..1..0..1 %e A298659 ..0..0..1..0..1..0..1. .1..0..1..0..1..0..1. .0..0..0..1..1..0..0 %Y A298659 Cf. A298660. %K A298659 nonn,new %O A298659 1,1 %A A298659 _R. H. Hardin_, Jan 24 2018 %I A298658 %S A298658 5,49,34,173,431,1116,2562,6711,17405,48462,125671,334571,901387, %T A298658 2398458,6398293,17158639,45911441,122885609,329214517,881849126, %U A298658 2361859125,6328533398,16955926709,45429854362,121730159805,326179436515 %N A298658 Number of nX6 0..1 arrays with every element equal to 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298658 Column 6 of A298660. %H A298658 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298658 Some solutions for n=5 %e A298658 ..0..1..0..0..1..1. .0..1..0..1..1..0. .0..1..1..1..1..0. .0..1..0..0..1..0 %e A298658 ..1..0..0..0..0..0. .1..0..1..0..0..1. .0..0..1..1..0..0. .1..0..1..1..1..0 %e A298658 ..0..0..0..1..0..1. .1..0..0..0..1..0. .0..0..1..1..1..1. .0..0..1..1..1..0 %e A298658 ..0..0..0..1..1..0. .1..0..0..0..0..1. .1..0..1..0..1..0. .0..0..1..0..1..0 %e A298658 ..0..1..1..0..1..0. .1..0..1..1..1..0. .1..0..0..0..1..0. .0..1..1..0..1..1 %Y A298658 Cf. A298660. %K A298658 nonn,new %O A298658 1,1 %A A298658 _R. H. Hardin_, Jan 24 2018 %I A298657 %S A298657 3,23,23,85,177,431,876,2137,5002,11687,27591,64253,150967,353682, %T A298657 830580,1954490,4575013,10742553,25221766,59150366,138829820, %U A298657 325959550,764905062,1795278490,4214144288,9890202526,23212413440,54483362577 %N A298657 Number of nX5 0..1 arrays with every element equal to 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298657 Column 5 of A298660. %H A298657 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298657 Some solutions for n=5 %e A298657 ..0..1..0..1..1. .0..0..0..1..0. .0..0..0..1..1. .0..1..1..1..0 %e A298657 ..1..0..1..0..0. .1..0..0..0..1. .1..0..0..0..0. .1..0..0..0..1 %e A298657 ..0..1..0..1..0. .1..0..0..0..0. .1..0..0..1..1. .1..0..0..1..0 %e A298657 ..1..0..1..0..0. .1..0..1..0..0. .0..0..0..1..0. .0..0..0..0..1 %e A298657 ..0..1..0..1..1. .0..1..0..1..0. .1..1..1..0..1. .1..1..1..1..0 %Y A298657 Cf. A298660. %K A298657 nonn,new %O A298657 1,1 %A A298657 _R. H. Hardin_, Jan 24 2018 %I A298656 %S A298656 2,13,19,40,85,173,322,635,1325,2806,5877,12293,25318,52348,110032, %T A298656 230666,481721,1008645,2105418,4397869,9221888,19306816,40379476, %U A298656 84574182,176999321,370432095,776067635,1625005774,3401774504,7125184125 %N A298656 Number of nX4 0..1 arrays with every element equal to 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298656 Column 4 of A298660. %H A298656 R. H. Hardin, Table of n, a(n) for n = 1..210 %F A298656 Empirical: a(n) = 3*a(n-1) +a(n-3) -14*a(n-4) -a(n-5) +33*a(n-6) -58*a(n-7) +8*a(n-8) +31*a(n-9) +257*a(n-10) -83*a(n-11) -415*a(n-12) +258*a(n-13) +257*a(n-14) -701*a(n-15) -1280*a(n-16) +817*a(n-17) +2138*a(n-18) -1277*a(n-19) -1528*a(n-20) +3841*a(n-21) +4395*a(n-22) -4674*a(n-23) -3840*a(n-24) +6126*a(n-25) +1522*a(n-26) -10670*a(n-27) -7803*a(n-28) +11150*a(n-29) +1242*a(n-30) -18501*a(n-31) +5389*a(n-32) +19940*a(n-33) -6561*a(n-34) -14251*a(n-35) +20541*a(n-36) +18531*a(n-37) -24833*a(n-38) -10425*a(n-39) +29434*a(n-40) +13173*a(n-41) -23604*a(n-42) -9671*a(n-43) +14824*a(n-44) -2310*a(n-45) -14701*a(n-46) -6124*a(n-47) +4664*a(n-48) +2279*a(n-49) -4344*a(n-50) +3221*a(n-51) +5148*a(n-52) -1564*a(n-53) -1655*a(n-54) +410*a(n-55) -29*a(n-56) +385*a(n-57) +1679*a(n-58) +770*a(n-59) -1293*a(n-60) -895*a(n-61) +499*a(n-62) +423*a(n-63) -72*a(n-64) -174*a(n-65) -69*a(n-66) +20*a(n-67) +16*a(n-68) +10*a(n-69) +5*a(n-70) -2*a(n-71) -a(n-72) for n>73 %e A298656 Some solutions for n=5 %e A298656 ..0..1..1..0. .0..1..1..0. .0..0..1..1. .0..1..1..1. .0..1..0..1 %e A298656 ..0..0..0..1. .0..0..0..0. .1..0..1..0. .0..0..0..0. .0..1..1..0 %e A298656 ..0..0..0..1. .0..0..0..0. .1..1..1..0. .0..0..0..1. .0..1..1..1 %e A298656 ..0..1..0..1. .1..0..1..0. .1..1..1..0. .1..0..0..1. .1..1..1..1 %e A298656 ..1..1..0..0. .1..0..1..1. .1..0..0..1. .0..1..0..1. .0..0..0..1 %Y A298656 Cf. A298660. %K A298656 nonn,new %O A298656 1,1 %A A298656 _R. H. Hardin_, Jan 24 2018 %I A298655 %S A298655 0,3,15,40,177,1116,7964,89543,1367704,32451809,1028816587 %N A298655 Number of nXn 0..1 arrays with every element equal to 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298655 Diagonal of A298660. %e A298655 Some solutions for n=5 %e A298655 ..0..1..0..0..1. .0..0..0..0..0. .0..1..1..1..1. .0..0..0..0..1 %e A298655 ..0..1..1..1..0. .1..1..1..1..1. .0..0..1..1..0. .1..1..1..1..1 %e A298655 ..0..1..1..1..0. .0..0..0..0..0. .1..1..1..1..0. .0..1..1..0..0 %e A298655 ..0..1..0..1..0. .1..1..1..1..1. .1..0..0..0..1. .0..1..1..1..1 %e A298655 ..1..0..1..0..1. .0..0..0..0..0. .0..0..0..1..0. .1..1..1..0..0 %Y A298655 Cf. A298660. %K A298655 nonn,new %O A298655 1,2 %A A298655 _R. H. Hardin_, Jan 24 2018 %I A298653 %S A298653 0,1,1,1,4,1,2,17,17,2,3,49,48,49,3,5,166,146,146,166,5,8,573,424,466, %T A298653 424,573,8,13,1933,1274,1446,1446,1274,1933,13,21,6538,3820,4648,5124, %U A298653 4648,3820,6538,21,34,22165,11529,14888,18271,18271,14888,11529,22165,34 %N A298653 T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 3, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298653 Table starts %C A298653 ..0.....1.....1......2......3.......5........8........13.........21.........34 %C A298653 ..1.....4....17.....49....166.....573.....1933......6538......22165......75089 %C A298653 ..1....17....48....146....424....1274.....3820.....11529......34783.....104826 %C A298653 ..2....49...146....466...1446....4648....14888.....47399.....150849.....480015 %C A298653 ..3...166...424...1446...5124...18271....62544....215035.....739962....2537660 %C A298653 ..5...573..1274...4648..18271...75562...291784...1142188....4518674...17656883 %C A298653 ..8..1933..3820..14888..62544..291784..1277500...5758443...26328879..118276552 %C A298653 .13..6538.11529..47399.215035.1142188..5758443..30337435..163931288..865848465 %C A298653 .21.22165.34783.150849.739962.4518674.26328879.163931288.1060912744.6689007779 %H A298653 R. H. Hardin, Table of n, a(n) for n = 1..287 %F A298653 Empirical for column k: %F A298653 k=1: a(n) = a(n-1) +a(n-2) %F A298653 k=2: a(n) = 3*a(n-1) +a(n-2) +2*a(n-3) -2*a(n-4) -4*a(n-5) for n>6 %F A298653 k=3: [order 11] for n>13 %F A298653 k=4: [order 24] for n>27 %e A298653 Some solutions for n=5 k=4 %e A298653 ..0..0..0..1. .0..0..1..0. .0..1..0..0. .0..1..1..0. .0..0..1..0 %e A298653 ..1..1..1..0. .0..1..0..1. .1..0..1..0. .1..0..0..0. .1..0..1..0 %e A298653 ..1..0..0..1. .1..1..0..1. .1..0..1..1. .0..1..1..1. .1..1..1..0 %e A298653 ..0..1..1..1. .0..1..0..1. .1..0..0..0. .0..0..0..1. .1..0..1..0 %e A298653 ..0..0..0..0. .0..0..1..1. .0..1..1..1. .1..1..1..0. .1..0..1..0 %Y A298653 Column 1 is A000045(n-1). %Y A298653 Column 2 is A297817. %Y A298653 Column 3 is A297988. %Y A298653 Column 4 is A297989. %K A298653 nonn,tabl,new %O A298653 1,5 %A A298653 _R. H. Hardin_, Jan 24 2018 %I A298652 %S A298652 8,1933,3820,14888,62544,291784,1277500,5758443,26328879,118276552, %T A298652 531491667,2406395782,10892034473,49246533056,222848686116, %U A298652 1008667256385,4564536675139,20660407068626,93538032416560,423503341613182 %N A298652 Number of nX7 0..1 arrays with every element equal to 1, 2, 3, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298652 Column 7 of A298653. %H A298652 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298652 Some solutions for n=5 %e A298652 ..0..0..0..1..0..1..0. .0..0..1..0..1..0..0. .0..1..0..1..0..1..1 %e A298652 ..0..1..1..0..0..0..1. .0..1..0..1..1..1..1. .1..0..0..0..1..0..1 %e A298652 ..0..1..0..0..0..0..0. .0..1..1..1..1..1..1. .0..0..0..0..0..0..1 %e A298652 ..0..1..0..1..0..0..1. .0..0..1..1..1..0..0. .0..1..0..0..1..0..1 %e A298652 ..0..0..1..1..0..0..1. .1..1..0..1..0..1..1. .0..1..0..0..1..0..1 %Y A298652 Cf. A298653. %K A298652 nonn,new %O A298652 1,1 %A A298652 _R. H. Hardin_, Jan 24 2018 %I A298651 %S A298651 5,573,1274,4648,18271,75562,291784,1142188,4518674,17656883,69095683, %T A298651 271751608,1068599426,4200796398,16523613182,65016945420,255906018679, %U A298651 1007618786294,3968688229186,15635494589187,61614746058306 %N A298651 Number of nX6 0..1 arrays with every element equal to 1, 2, 3, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298651 Column 6 of A298653. %H A298651 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298651 Some solutions for n=5 %e A298651 ..0..1..0..1..0..1. .0..1..0..1..0..1. .0..1..0..0..1..0. .0..0..1..1..0..0 %e A298651 ..1..0..1..1..1..0. .1..0..0..0..1..1. .0..1..0..0..1..0. .1..0..1..1..0..1 %e A298651 ..1..1..1..1..1..1. .0..0..0..0..0..0. .0..0..0..0..0..0. .1..1..1..1..1..1 %e A298651 ..1..0..1..1..0..1. .0..1..0..0..0..0. .0..1..0..0..0..1. .0..1..1..1..0..1 %e A298651 ..0..1..0..1..1..0. .1..0..1..0..1..1. .1..0..1..0..1..0. .1..0..1..0..1..0 %Y A298651 Cf. A298653. %K A298651 nonn,new %O A298651 1,1 %A A298651 _R. H. Hardin_, Jan 24 2018 %I A298650 %S A298650 3,166,424,1446,5124,18271,62544,215035,739962,2537660,8744241, %T A298650 30188614,104165837,359665578,1242411171,4292458468,14835586385, %U A298650 51288707980,177340234374,613288580624,2121228689542,7337686991133,25384797504647 %N A298650 Number of nX5 0..1 arrays with every element equal to 1, 2, 3, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298650 Column 5 of A298653. %H A298650 R. H. Hardin, Table of n, a(n) for n = 1..210 %e A298650 Some solutions for n=5 %e A298650 ..0..1..0..1..0. .0..1..0..1..0. .0..1..1..0..1. .0..0..1..0..1 %e A298650 ..1..0..0..0..1. .1..0..0..0..1. .0..0..1..1..0. .1..1..1..1..0 %e A298650 ..0..0..0..0..0. .0..0..0..0..0. .1..1..1..1..1. .1..1..1..1..1 %e A298650 ..1..0..0..1..0. .1..0..0..1..0. .1..1..1..1..0. .0..0..1..1..0 %e A298650 ..0..1..0..0..1. .1..0..0..1..1. .0..0..1..0..1. .1..1..1..0..1 %Y A298650 Cf. A298653. %K A298650 nonn,new %O A298650 1,1 %A A298650 _R. H. Hardin_, Jan 24 2018 %I A298649 %S A298649 0,4,48,466,5124,75562,1277500,30337435,1060912744,50167771709, %T A298649 3415576723102,354153850821427 %N A298649 Number of nXn 0..1 arrays with every element equal to 1, 2, 3, 6 or 7 king-move adjacent elements, with upper left element zero. %C A298649 Diagonal of A298653. %e A298649 Some solutions for n=5 %e A298649 ..0..1..1..0..1. .0..0..1..1..0. .0..0..1..0..1. .0..1..1..0..0 %e A298649 ..0..0..1..1..0. .1..0..1..1..0. .1..1..1..1..0. .0..1..1..0..0 %e A298649 ..1..1..1..1..1. .1..1..1..1..1. .1..1..1..1..1. .1..1..1..1..1 %e A298649 ..1..1..1..1..0. .0..1..1..1..0. .0..0..1..1..0. .0..1..1..1..1 %e A298649 ..0..0..1..0..1. .1..0..1..0..1. .0..1..1..0..1. .1..0..1..0..0 %Y A298649 Cf. A298653. %K A298649 nonn,new %O A298649 1,2 %A A298649 _R. H. Hardin_, Jan 24 2018 %I A297867 %S A297867 776151559,3518958160000 %N A297867 3-powerful numbers that can be written as the sum of two coprime 3-powerful numbers. %C A297867 Any counterexample to the Beal conjecture, i.e., the statement that the diophantine equation A^x + B^y = C^z has no solution for pairwise coprime A, B, C and x, y, z > 2, has to be a term from this sequence. %H A297867 Wikipedia, Beal conjecture %e A297867 3518958160000 = 1392672604221 + 2126285555779 = 3^4 * 29^3 * 89^3 + 7^3 * 11^3 * 167^3 = 2^7 * 5^4 * 353^3. %o A297867 (PARI) is_a036966(n) = my(e=factor(n)[, 2]~); if(#e==0 || vecmin(e) < 3, return(0)); 1 %o A297867 is(n) = if(!is_a036966(n), return(0)); my(x=1, y=n-1); while(x < y, if(gcd(x, y)==1 && n==x+y && is_a036966(x) && is_a036966(y), return(1)); x++; y--); 0 %Y A297867 Cf. A036966. %K A297867 nonn,hard,bref,more,new %O A297867 1,1 %A A297867 _Felix Fröhlich_, Jan 07 2018 %I A297850 %S A297850 2,2,2,2,2,2,3,2,2,2,3,3,2,3,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,2, %T A297850 2,2,2,3,2,2,3,2,2,2,2,3,2,3,2,3,2,2,2,2,2,3,2,2,2,2,2,2,2,2,2,2,2,2, %U A297850 2,2 %N A297850 Least common prime factor of the members of n-th amicable pair, or 0 if the two members of the pair are coprime. %C A297850 The question whether a(n) = 0 for any n is an open problem. %C A297850 This is different from A171092 (cf. Chernykh link). %C A297850 If a(n) = 0, then A001221(A259180(2*n+1)*A259180(2*n+2)) > 21 (cf. Hagis, 1975). %H A297850 S. Chernykh, Amicable pairs news %H A297850 P. Hagis, Jr., On the Number of Prime Factors of a Pair of Relatively Prime Amicable Numbers, Mathematics Magazine, Vol. 48, No. 5 (1975), 263-266. %H A297850 Wikipedia, Amicable numbers %F A297850 a(n) = A297934(A259180(2*n+2), A259180(2*n+1)). %Y A297850 Cf. A002025, A122967, A171092, A259180, A297934. %K A297850 nonn,more,new %O A297850 0,1 %A A297850 _Felix Fröhlich_, Jan 10 2018 %I A297849 %S A297849 65,145,217,325,485,561,721,785,901,904,1025,1105,1157,1261,1281,1333, %T A297849 1445,1729,1765,1905,1937,2117,2305,2465,2501,2701,2705,3126,3201, %U A297849 3365,3421,3565,3601,3845,4033,4097,4369,4625,4901,5185,5777,5833,6085,6401,6499 %N A297849 Composites c where another composite d < c exists such that c and d satisfy c^(d-1) == 1 (mod d^2) and d^(c-1) == 1 (mod c) or satisfy c^(d-1) == 1 (mod d) and d^(c-1) == 1 (mod c^2). %C A297849 Are there any composites c where a composite d with d < c exists such that both c^(d-1) == 1 (mod d^2) and d^(c-1) == 1 (mod c^2)? %e A297849 The composites 8 and 65 satisfy the congruences 65^(8-1) == 1 (mod 8^2) and 8^(65-1) == 1 (mod 65), so 65 is a term of the sequence. %t A297849 With[{s = Select[Range@ 3000, CompositeQ]}, Select[s, Function[c, AnyTrue[Take[s, First@ FirstPosition[s, c]], Or[And[PowerMod[c, (# - 1), #^2] == 1, PowerMod[#, (c - 1), c] == 1], And[PowerMod[c, (# - 1), #] == 1, PowerMod[#, (c - 1), c^2] == 1]] &]]]] (* _Michael De Vlieger_, Jan 11 2018 *) %o A297849 (PARI) is(n) = forcomposite(c=1, n-1, if((Mod(n, c^2)^(c-1)==1 && Mod(c, n)^(n-1)==1) || (Mod(n, c)^(c-1)==1 && Mod(c, n^2)^(n-1)==1), return(1))); 0 %o A297849 forcomposite(c=1, , if(is(c), print1(c, ", "))) %Y A297849 Subsequence of A270574. %K A297849 nonn,new %O A297849 1,1 %A A297849 _Felix Fröhlich_, Jan 07 2018 %I A297848 %S A297848 381348997,717636389,778090129,1496216791 %N A297848 Irregular primes with irregularity index 8. %H A297848 W. Hart, D. Harvey and W. Ong, Irregular primes to two billion, arXiv:1605.02398 [math.NT], 2016. %H A297848 W. Hart, D. Harvey and W. Ong, Irregular primes to two billion, Mathematics of Computation, 86 (2017), 3031-3049. %Y A297848 Cf. A073276, A073277, A060975, A219332, A219333, A219334, A219335. %K A297848 nonn,hard,more,new %O A297848 1,1 %A A297848 _Felix Fröhlich_, Jan 07 2018 %I A297839 %S A297839 1,3,4,14,18,23,62,95,423,5339,12352,108359,129805,5334194,82007322, %T A297839 90401717,199671691,434184265,655956850 %N A297839 Numbers k > 0 that set a new record for the closeness of (4/3)*Pi*k^3 to an integer. %C A297839 Integer radii such that the volume of the corresponding sphere is closer to an integer than for any smaller integer radius. %e A297839 k | (4/3)*Pi*k^3 | Deviation from integer %e A297839 --------------------------------------------------------------------------- %e A297839 1 | 4.188790204786390... | 0.188790204786390... %e A297839 3 | 113.097335529232556... | 0.097335529232556... %e A297839 4 | 268.082573106329023... | 0.082573106329023... %e A297839 14 | 11494.040321933856861... | 0.040321933856861... %e A297839 18 | 24429.024474314232222... | 0.024474314232222... %e A297839 23 | 50965.010421636019109... | 0.010421636019109... %e A297839 62 | 998305.991926330990581... | 0.008073669009418... %e A297839 95 | 3591364.001828731970435... | 0.001828731970435... %e A297839 423 | 317036825.999590816501793... | 0.000409183498206... %e A297839 5339 | 637482653747.999839504336479... | 0.000160495663520... %e A297839 12352 | 7894060641354.000003942767448... | 0.000003942767448... %e A297839 108359 | 5329464512150064.999997849950689... | 0.000000215004931... %e A297839 129805 | 9161421693208264.000000035388795... | 0.000000035388795... %e A297839 5334194 | 635762677398025211698.999999995151941... | 0.000000004848058... %o A297839 (PARI) closeness(n) = my(v=(4/3)*Pi*n^3); if(round(v) > v, return(round(v)-v), return(v-round(v))) %o A297839 my(r=1, k=1, c=0); while(1, c=closeness(k); if(c < r, print1(k, ", "); r=c); k++) %Y A297839 Cf. A066645, A135973, A254714, A297840. %K A297839 nonn,hard,more,new %O A297839 1,2 %A A297839 _Felix Fröhlich_, Jan 07 2018 %E A297839 a(15)-a(19) from _Jon E. Schoenfield_, Jan 07 2018 %I A298034 %S A298034 1,7,19,43,73,115,163,223,289,367,451,547,649,763,883,1015,1153,1303, %T A298034 1459,1627,1801,1987,2179,2383,2593,2815,3043,3283,3529,3787,4051, %U A298034 4327,4609,4903,5203,5515,5833,6163,6499,6847,7201,7567,7939,8323,8713,9115,9523,9943,10369,10807,11251,11707 %N A298034 Partial sums of A298033. %H A298034 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298034 Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1). %F A298034 G.f.: (1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)^3*(1 + x)). %F A298034 From _Colin Barker_, Jan 25 2018: (Start) %F A298034 a(n) = (9*n^2 + 2) / 2 for n even. %F A298034 a(n) = (9*n^2 + 5) / 2 for n odd. %F A298034 a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>3. (End) %F A298034 a(4*k+r) = 36*k*(2*k + r) + a(r) for r = 0..3. Example: if n=29 then k=7 and r=1, hence a(29) = 36*7*(2*7 + 1) + 7 = 3787. - _Bruno Berselli_, Jan 25 2018 %o A298034 (PARI) Vec((1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)^3*(1 + x)) + O(x^50)) \\ _Colin Barker_, Jan 25 2018 %Y A298034 Cf. A298033. %K A298034 nonn,easy,new %O A298034 0,2 %A A298034 _N. J. A. Sloane_, Jan 21 2018, corrected Jan 24 2018 %I A298641 %S A298641 1,0,1,1,2,1,8,6,45,100,377,1181,4063,13225,45218,150928,511970, %T A298641 1717140,5777895,19308880,64360153,213446697,705095144,2317573307, %U A298641 7583418322,24690176885,80003762726,257959340058,827713115396,2642967441892,8398644246488 %N A298641 Number of partitions of n^3 into cubes > 1. %H A298641 Vaclav Kotesovec, Table of n, a(n) for n = 0..100 %H A298641 Index entries for sequences related to sums of cubes %H A298641 Index entries for related partition-counting sequences %F A298641 a(n) = [x^(n^3)] Product_{k>=2} 1/(1 - x^(k^3)). %F A298641 a(n) = A078128(A000578(n)). %F A298641 a(n) ~ exp(4*(Gamma(1/3) * Zeta(4/3))^(3/4) * n^(3/4) / 3^(3/2)) * (Gamma(1/3) * Zeta(4/3))^(3/2) / (8 * 3^(5/2) * Pi^2 * n^6). - _Vaclav Kotesovec_, Jan 31 2018 %e A298641 a(4) = 2 because we have [64] and [8, 8, 8, 8, 8, 8, 8, 8]. %p A298641 g:= proc(n, L) # number of partitions of n into cubes > 1 and <= L %p A298641 option remember; %p A298641 local t,k; %p A298641 t:= 0; %p A298641 if n = 0 then return 1 fi; %p A298641 if n < 8 then return 0 fi; %p A298641 for k from 2 while k^3 <= min(n,L) do %p A298641 t:= t + procname(n-k^3, k^3) %p A298641 od %p A298641 end proc: %p A298641 f:= n -> g(n^3, n^3): %p A298641 map(f, [$0..50]); # _Robert Israel_, Jan 24 2018 %t A298641 mx = 30; s = Series[Product[1/(1 - x^(k^3)), {k, 2, mx}], {x, 0, mx^3}]; Table[ CoefficientList[s, x][[1 + n^3]], {n, 0, mx}] (* _Robert G. Wilson v_, Jan 24 2018 *) %Y A298641 Cf. A000578, A003108, A030272, A078128, A092362, A259792, A279329, A280130, A290247. %K A298641 nonn,new %O A298641 0,5 %A A298641 _Ilya Gutkovskiy_, Jan 24 2018 %I A298640 %S A298640 1,0,1,1,2,8,12,129,874,9630,167001,3043147,72844510,2423789655, %T A298640 106665874384,6156805673648,470151743582651,47558937432498729, %U A298640 6363358599941131580,1126147544855148769425,263646401550138303553708,81649922556593759124887197 %N A298640 Number of compositions (ordered partitions) of n^2 into squares > 1. %H A298640 Alois P. Heinz, Table of n, a(n) for n = 0..128 %H A298640 Index entries for sequences related to sums of squares %H A298640 Index entries for sequences related to compositions %F A298640 a(n) = [x^(n^2)] 1/(1 - Sum_{k>=2} x^(k^2)). %F A298640 a(n) = A280542(A000290(n)). %e A298640 a(5) = 8 because we have [25], [16, 9], [9, 16], [9, 4, 4, 4, 4], [4, 9, 4, 4, 4], [4, 4, 9, 4, 4], [4, 4, 4, 9, 4] and [4, 4, 4, 4, 9]. %p A298640 b:= proc(n) option remember; `if`(n=0, 1, %p A298640 add(b(n-j^2), j=2..isqrt(n))) %p A298640 end: %p A298640 a:= n-> b(n^2): %p A298640 seq(a(n), n=0..25); # _Alois P. Heinz_, Feb 05 2018 %Y A298640 Cf. A000290, A006456, A078134, A224366, A280542, A298642. %K A298640 nonn,new %O A298640 0,5 %A A298640 _Ilya Gutkovskiy_, Jan 24 2018 %I A298643 %S A298643 11,191,2,223,5,2,227,7,3,2,2111,17,7,3,2,3847,31,13,7,3,2,229631,41, %T A298643 23,11,5,3,2,246271,53,29,13,11,5,3,2,262111,157,47,17,31,11,5,3,2, %U A298643 786431,229,53,19,47,13,7,5,3,2,1046527,239,101,23,71,17,13,7,5 %N A298643 Array A(n, k) read by antidiagonals downwards: k-th base-n non-repunit prime p such that all numbers resulting from switching any two adjacent digits in the base-n representation of p are prime, where k runs over the positive integers, i.e., the offset of k is 1. %C A298643 Conjecture: All rows of the array are infinite. %C A298643 If the above conjecture is false, then this should have keyword "tabf" rather than "tabl". %C A298643 Row n is a supersequence of the base-n non-repunit absolute primes. For example, row 10 (A107845) is a supersequence of the decimal non-repunit absolute primes (A129338). %e A298643 The base-3 representation of 251 is 100022. Base-3 numbers that can be obtained by switching any two adjacent base-3 digits are 10022 and 100202. These two numbers are 89 and 263, respectively, when converted to decimal, and both 89 and 263 are prime. Since 251 is the 12-th number with this property in base 3, A(3, 12) = 251. %e A298643 Array starts %e A298643 11, 191, 223, 227, 2111, 3847, 229631, 246271, 262111, 786431, 1046527, 1047551 %e A298643 2, 5, 7, 17, 31, 41, 53, 157, 229, 239, 241, 251 %e A298643 2, 3, 7, 13, 23, 29, 47, 53, 101, 127, 149, 151 %e A298643 2, 3, 7, 11, 13, 17, 19, 23, 43, 131, 281, 311 %e A298643 2, 3, 5, 11, 31, 47, 71, 83, 103, 107, 151, 191 %e A298643 2, 3, 5, 11, 13, 17, 19, 23, 29, 37, 41, 43 %e A298643 2, 3, 5, 7, 13, 29, 31, 41, 43, 47, 59, 61 %e A298643 2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 43, 47 %e A298643 2, 3, 5, 7, 13, 17, 31, 37, 71, 73, 79, 97 %e A298643 2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 43 %e A298643 2, 3, 5, 7, 11, 17, 61, 67, 71, 89, 137, 163 %o A298643 (PARI) switchdigits(v, pos) = my(vt=v[pos]); v[pos]=v[pos+1]; v[pos+1]=vt; v %o A298643 decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i]) %o A298643 is(p, base) = my(db=digits(p, base)); if(vecmin(db)==1 && vecmax(db)==1, return(0)); for(k=1, #db-1, my(x=decimal(switchdigits(db, k), base)); if(!ispseudoprime(x), return(0))); 1 %o A298643 array(n, k) = for(x=2, n+1, my(i=0); forprime(p=1, , if(is(p, x), print1(p, ", "); i++); if(i==k, print(""); break))) %o A298643 array(6, 10) \\ print initial 6 rows and 10 columns of array %Y A298643 Cf. A107845 (row 10), A129338. %K A298643 nonn,tabl,base,new %O A298643 2,1 %A A298643 _Felix Fröhlich_, Jan 24 2018 %I A298642 %S A298642 1,0,1,1,1,2,1,2,1,2,2,2,1,5,2,10,4,12,12,11,19,23,43,50,55,78,120, %T A298642 126,234,207,407,385,701,712,1090,1231,1850,2102,3054,3385,4988,5584, %U A298642 7985,9746,12205,15737,18968,25157,30927,39043,47708,61915,74592,99554 %N A298642 Number of partitions of n^2 into distinct squares > 1. %H A298642 Index entries for sequences related to sums of squares %H A298642 Index entries for related partition-counting sequences %F A298642 a(n) = [x^(n^2)] Product_{k>=2} (1 + x^(k^2)). %F A298642 a(n) = A280129(A000290(n)). %e A298642 a(5) = 2 because we have [25] and [16, 9]. %Y A298642 Cf. A000290, A001156, A030273, A033461, A037444, A078134, A092362, A093115, A093116, A280129, A298640. %K A298642 nonn,new %O A298642 0,6 %A A298642 _Ilya Gutkovskiy_, Jan 24 2018 %I A298033 %S A298033 1,6,12,24,30,42,48,60,66,78,84,96,102,114,120,132,138,150,156,168, %T A298033 174,186,192,204,210,222,228,240,246,258,264,276,282,294,300,312,318, %U A298033 330,336,348,354,366,372,384,390,402,408,420,426,438,444,456,462,474,480,492,498,510,516,528,534,546,552 %N A298033 Coordination sequence of the Dual(3.4.6.4) tiling with respect to a hexavalent node. %C A298033 Also known as the mta net. %C A298033 This is one of the Laves tilings. %D A298033 Chaim Goodman-Strauss and N. J. A. Sloane, The Coloring Book Approach to Finding Coordination Sequences, 2018 [will be added here soon] %H A298033 Colin Barker, Table of n, a(n) for n = 0..1000 %H A298033 Reticular Chemistry Structure Resource (RCSR), The mta tiling (or net) %H A298033 N. J. A. Sloane, The Dual(3.4.6.4) tiling %H A298033 N. J. A. Sloane, The subgraph H shown in one 60-degree sector of the graph of the tiling. %H A298033 N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database] %H A298033 Index entries for linear recurrences with constant coefficients, signature (1,1,-1). %F A298033 Theorem: For n>0, a(n) = 9*n-6 if n is even, a(n) = 9*n-3 if n is odd. %F A298033 The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links. %F A298033 G.f.: (1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)*(1 - x^2)). %F A298033 First differences are 1, 5, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, ... %F A298033 a(n) = a(n-1) + a(n-2) - a(n-3) for n>3. - _Colin Barker_, Jan 25 2018 %p A298033 f6:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 9*n-6 else 9*n-3; fi; end; %p A298033 [seq(f6(n),n=0..80)]; %o A298033 (PARI) Vec((1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ _Colin Barker_, Jan 25 2018 %Y A298033 List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458. %Y A298033 Cf. A008574, A038764 (partial sums), A298029 (coordination sequence for a trivalent node), A298031 (coordination sequence for a tetravalent node). %K A298033 nonn,easy,new %O A298033 0,2 %A A298033 _N. J. A. Sloane_, Jan 21 2018, corrected Jan 24 2018. %I A285188 %S A285188 0,4,13,45,95,203,350,606,930,1430,2035,2899,3913,5285,6860,8908, %T A285188 11220,14136,17385,21385,25795,31119,36938,43850,51350,60138,69615, %U A285188 80591,92365,105865,120280,136664,154088,173740,194565,217893,242535,269971 %N A285188 a(n) = Sum_{k=1..n} (k^2*floor(k/2)). %H A285188 Robert Israel, Table of n, a(n) for n = 1..10000 %H A285188 Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1). %F A285188 Theorem: a(n) = (1/8)*n^2*(n+1)^2 - (2/3)*floor((n+1)/2)^3 + (1/6)*floor((n+1)/2). %F A285188 From _Chai Wah Wu_, Apr 24 2017: (Start) %F A285188 a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n > 8. %F A285188 G.f.: x^2*(x^4 + 3*x^3 + 11*x^2 + 5*x + 4)/((1 - x)^5*(1 + x)^3). (End) %F A285188 a(n) = n*(n+1)*(3*n^2+n-1+3*(-1)^n)/24. - _Robert Israel_, Apr 26 2017 %e A285188 For n = 4, a(4) = 1^2*floor(1/2) + 2^2*floor(2/2) + 3^2*floor(3/2) + 4^2*floor(4/2) = 0 + 4 + 9 + 32 = 45. %p A285188 seq( n*(n+1)*(3*n^2+n-1+3*(-1)^n)/24, n=1..100); # _Robert Israel_, Apr 26 2017 %o A285188 (MATLAB) s = @(n) sum((1:n).^2.*floor((1:n)/2)); %summation handle function %o A285188 s_cf = @(n) 1/8*n^2*(n+1)^2 - 2/3*floor((n+1)/2)^3 + 1/6*floor((n+1)/2); %faster closed-form handle function %o A285188 (PARI) a(n) = sum(k=1, n, k^2*(k\2)); \\ _Michel Marcus_, Apr 24 2017 %Y A285188 Cf. A049779. %Y A285188 Parial sums of A265645. %K A285188 new,nonn %O A285188 1,2 %A A285188 _Néstor Jofré_, Apr 24 2017 %I A283791 %S A283791 449,4159,4801,4999,8191,11551,11969,15731,16561,22541,26449,28729, %T A283791 31249,33857,35153,38501,39929,42283,45631,46817,47431,47501,48049, %U A283791 51679,52021,62929,63799,68449,69191,81919,83231,84967,89909,94771,97499,100049,104059 %N A283791 List of prime numbers p such that all prime factors of p+1 and p-1 are smaller than the cube root of p. %H A283791 Lei Zhou, Table of n, a(n) for n = 1..10000 %e A283791 449 is a prime number. 449+1 = 450 = 2*3^2*5^2, 5^3 = 125 < 449; 449-1 = 448 = 2^6*7, 7^3 = 343 < 449, so 449 is in this list. %e A283791 457 is a prime number. 457+1 = 458 = 2*229, 229^3 > 457, so 457 is NOT in this list. %p A283791 with(numtheory): P:=proc(n) local a,b,k,ok; a:=ithprime(n); %p A283791 b:=[op(factorset(a-1) union factorset(a+1))]; ok:=1; %p A283791 for k from 1 to nops(b) do if b[k]>evalf(a^(1/3)) then ok:=0; %p A283791 break; fi; od; if ok=1 then a; fi; end: seq(P(i),i=1..10^4); %p A283791 # _Paolo P. Lava_, Nov 10 2017 %t A283791 p = 1; Table[ %t A283791 While[p = NextPrime[p]; fp = Last[FactorInteger[p + 1]][[1]]; %t A283791 fm = Last[FactorInteger[p - 1]][[1]]; (fp^3 >= p) || (fm^3 >= %t A283791 p)]; p, {n, 1, 37}] %o A283791 (PARI) isok(p) = isprime(p) && (p>2) && (vecmax(factor(p-1)[,1])^3 < p) && (vecmax(factor(p+1)[,1])^3 < p); \\ _Michel Marcus_, Jan 10 2018 %Y A283791 Cf. A000040. %K A283791 new,nonn,easy %O A283791 1,1 %A A283791 _Lei Zhou_, Mar 16 2017 %E A283791 Definition corrected by _Zak Seidov_, Dec 04 2017