|
|
A195162
|
|
Generalized 12-gonal numbers: k*(5*k-4) for k = 0, +-1, +-2, ...
|
|
51
|
|
|
0, 1, 9, 12, 28, 33, 57, 64, 96, 105, 145, 156, 204, 217, 273, 288, 352, 369, 441, 460, 540, 561, 649, 672, 768, 793, 897, 924, 1036, 1065, 1185, 1216, 1344, 1377, 1513, 1548, 1692, 1729, 1881, 1920, 2080, 2121, 2289, 2332, 2508, 2553, 2737, 2784, 2976, 3025
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
Also generalized dodecagonal numbers.
Also, numbers h such that 5*h + 4 is a square. - Bruno Berselli, Oct 10 2013
Exponents in expansion of Product_{n >= 1} (1 + x^(10*n-9))*(1 + x^(10*n-1))*(1 - x^(10*n)) = 1 + x + x^9 + x^12 + x^28 + .... - Peter Bala, Dec 10 2020
|
|
LINKS
|
|
|
FORMULA
|
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: x*(1+8*x+x^2)/((1+x)^2*(1-x)^3).
a(n) = (10*n*(n+1) + 3*(2*n+1)*(-1)^n - 3)/8.
a(n) = a(-n-1). (End)
Sum_{n>=1} 1/a(n) = (5 + 4*sqrt(1 + 2/sqrt(5))*Pi)/16. - Vaclav Kotesovec, Oct 05 2016
E.g.f.: (3*(1 - 2*x)*exp(-x) + (-3 +20*x +10*x^2)*exp(x))/8. - G. C. Greubel, Jul 04 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(5)/8 + sqrt(5)*log(phi)/4 - 5/16, where phi is the golden ratio (A001622). - Amiram Eldar, Feb 28 2022
|
|
MATHEMATICA
|
nn = 25; Sort[Table[n*(5*n - 4), {n, -nn, nn}]] (* T. D. Noe, Sep 23 2011 *)
|
|
PROG
|
(Magma) [0] cat &cat[[5*n^2-4*n, 5*n^2+4*n]: n in [1..25]]; // Vincenzo Librandi, Sep 26 2011
(PARI) vector(50, n, n--; (10*n^2 +10*n -3 +3*(-1)^n*(2*n+1))/8) \\ G. C. Greubel, Jul 04 2019
(Sage) [(10*n^2 +10*n -3 +3*(-1)^n*(2*n+1))/8 for n in (0..50)] # G. C. Greubel, Jul 04 2019
(GAP) List([0..50], n-> (10*n^2 +10*n -3 +3*(-1)^n*(2*n+1))/8) # G. C. Greubel, Jul 04 2019
|
|
CROSSREFS
|
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), this sequence (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|