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%I M1336 N0511 #552 May 25 2023 08:46:12
%S 0,1,2,5,7,12,15,22,26,35,40,51,57,70,77,92,100,117,126,145,155,176,
%T 187,210,222,247,260,287,301,330,345,376,392,425,442,477,495,532,551,
%U 590,610,651,672,715,737,782,805,852,876,925,950,1001,1027,1080,1107,1162,1190,1247,1276,1335
%N Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, ....
%C Partial sums of A026741. - _Jud McCranie_; corrected by _Omar E. Pol_, Jul 05 2012
%C From _R. K. Guy_, Dec 28 2005: (Start)
%C "Conway's relation twixt the triangular and pentagonal numbers: Divide the triangular numbers by 3 (when you can exactly):
%C 0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 ...
%C 0 - 1 2 .- .5 .7 .- 12 15 .- 22 26 .- .35 .40 .- ..51 ...
%C .....-.-.....+..+.....-..-.....+..+......-...-.......+....
%C "and you get the pentagonal numbers in pairs, one of positive rank and the other negative.
%C "Append signs according as the pair have the same (+) or opposite (-) parity.
%C "Then Euler's pentagonal number theorem is easy to remember:
%C "p(n-0) - p(n-1) - p(n-2) + p(n-5) + p(n-7) - p(n-12) - p(n-15) ++-- = 0^n
%C where p(n) is the partition function, the left side terminates before the argument becomes negative and 0^n = 1 if n = 0 and = 0 if n > 0.
%C "E.g. p(0) = 1, p(7) = p(7-1) + p(7-2) - p(7-5) - p(7-7) + 0^7 = 11 + 7 - 2 - 1 + 0 = 15."
%C (End)
%C The sequence may be used in order to compute sigma(n), as described in Euler's article. - _Thomas Baruchel_, Nov 19 2003
%C Number of levels in the partitions of n + 1 with parts in {1,2}.
%C a(n) is the number of 3 X 3 matrices (symmetrical about each diagonal) M = {{a, b, c}, {b, d, b}, {c, b, a}} such that a + b + c = b + d + b = n + 2, a,b,c,d natural numbers; example: a(3) = 5 because (a,b,c,d) = (2,2,1,1), (1,2,2,1), (1,1,3,3), (3,1,1,3), (2,1,2,3). - _Philippe Deléham_, Apr 11 2007
%C Also numbers a(n) such that 24*a(n) + 1 = (6*m - 1)^2 are odd squares: 1, 25, 49, 121, 169, 289, 361, ..., m = 0, +-1, +-2, ... . - _Zak Seidov_, Mar 08 2008
%C From _Matthew Vandermast_, Oct 28 2008: (Start)
%C Numbers n for which A000326(n) is a member of A000332. Cf. A145920.
%C This sequence contains all members of A000332 and all nonnegative members of A145919. For values of n such that n*(3*n - 1)/2 belongs to A000332, see A145919. (End)
%C Starting with offset 1 = row sums of triangle A168258. - _Gary W. Adamson_, Nov 21 2009
%C Starting with offset 1 = Triangle A101688 * [1, 2, 3, ...]. - _Gary W. Adamson_, Nov 27 2009
%C Starting with offset 1 can be considered the first in an infinite set generated from A026741. Refer to the array in A175005. - _Gary W. Adamson_, Apr 03 2010
%C Vertex number of a square spiral whose edges have length A026741. The two axes of the spiral forming an "X" are A000326 and A005449. The four semi-axes forming an "X" are A049452, A049453, A033570 and the numbers >= 2 of A033568. - _Omar E. Pol_, Sep 08 2011
%C A general formula for the generalized k-gonal numbers is given by n*((k - 2)*n - k + 4)/2, n=0, +-1, +-2, ..., k >= 5. - _Omar E. Pol_, Sep 15 2011
%C a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and 2*w = 2*x + y. - _Clark Kimberling_, Jun 04 2012
%C Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. - _Omar E. Pol_, Aug 04 2012
%C a(n) is the sum of the largest parts of the partitions of n+1 into exactly 2 parts. - _Wesley Ivan Hurt_, Jan 26 2013
%C Conway's relation mentioned by _R. K. Guy_ is a relation between triangular numbers and generalized pentagonal numbers, two sequences from different families, but as triangular numbers are also generalized hexagonal numbers in this case we have a relation between two sequences from the same family. - _Omar E. Pol_, Feb 01 2013
%C Start with the sequence of all 0's. Add n to each value of a(n) and the next n - 1 terms. The result is the generalized pentagonal numbers. - _Wesley Ivan Hurt_, Nov 03 2014
%C (6k + 1) | a(4k). (3k + 1) | a(4k+1). (3k + 2) | a(4k+2). (6k + 5) | a(4k+3). - _Jon Perry_, Nov 04 2014
%C Enge, Hart and Johansson proved: "Every generalised pentagonal number c >= 5 is the sum of a smaller one and twice a smaller one, that is, there are generalised pentagonal numbers a, b < c such that c = 2a + b." (see link theorem 5). - _Peter Luschny_, Aug 26 2016
%C The Enge, et al. result for c >= 5 also holds for c >= 2 if 0 is included as a generalized pentagonal number. That is, 2 = 2*1 + 0. - _Michael Somos_, Jun 02 2018
%C Suggestion for title, where n actually matches the list and b-file: "Generalized pentagonal numbers: k(n)*(3*k(n) - 1)/2, where k(n) = A001057(n) = [0, 1, -1, 2, -2, 3, -3, ...], n >= 0" - _Daniel Forgues_, Jun 09 2018 & Jun 12 2018
%C Generalized k-gonal numbers are the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (A005408) interleaved, with k >= 5. - _Omar E. Pol_, Jul 25 2018
%C The last digits form a symmetric cycle of length 40 [0, 1, 2, 5, ..., 5, 2, 1, 0], i.e., a(n) == a(n + 40) (mod 10) and a(n) == a(40*k - n - 1) (mod 10), 40*k > n. - _Alejandro J. Becerra Jr._, Aug 14 2018
%C Only 2, 5, and 7 are prime. All terms are of the form k*(k+1)/6, where 3 | k or 3 | k+1. For k > 6, the value divisible by 3 must have another factor d > 2, which will remain after the division by 6. - _Eric Snyder_, Jun 03 2022
%C 8*a(n) is the product of two even numbers one of which is n + n mod 2. - _Peter Luschny_, Jul 15 2022
%C a(n) is the dot product of [1, 2, 3...} and repeat[1, 1/2]. a(5) = 12 = [1, 2, 3, 4, 5] dot [1, 1/2, 1, 1/2, 1] = [1 + 1 + 3 + 2 + 5]. - _Gary W. Adamson_, Dec 10 2022
%C Every nonnegative number is the sum of four terms of this sequence [S. Realis]. - _N. J. A. Sloane_, May 07 2023
%D Enoch Haga, A strange sequence and a brilliant discovery, chapter 5 of Exploring prime numbers on your PC and the Internet, first revised ed., 2007 (and earlier ed.), pp. 53-70.
%D Ross Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 117.
%D Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, (to appear), section 7.2.1.4, equation (18).
%D Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, 2nd ed., Wiley, NY, 1966, p. 231.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001318/b001318.txt">Table of n, a(n) for n = 0..1000</a>
%H G. E. Andrews and J. A. Sellers, <a href="http://arxiv.org/abs/1401.5345">Congruences for the Fishburn Numbers</a>, arXiv preprint arXiv:1401.5345 [math.NT], 2014.
%H Paul Barry, <a href="http://arxiv.org/abs/1205.2565">On sequences with {-1, 0, 1} Hankel transforms</a>, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 18 2012
%H Burkard Polster (Mathologer), <a href="https://www.youtube.com/watch?v=iJ8pnCO0nTY">The hardest "What comes next?" (Euler's pentagonal formula)</a>, Youtube video, Oct 17 2020.
%H S. Cooper and M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00079-7">Results of Hurwitz type for three squares</a>, Discrete Math., Vol. 274, No. 1-3 (2004), pp. 9-24. See P(q).
%H Stephen Eberhart, <a href="/A001318/a001318_2.pdf">Letter to N. J. A. Sloane, Jan 19 1978</a>.
%H John Elias, <a href="/A001318/a001318.png">Illustration of Initial Terms: Generalized Penthexagrams</a>
%H John Elias, <a href="/A001318/a001318_1.png">Illustration: Star Number Fractal</a>
%H John Elias, <a href="/A001318/a001318_2.png">Illustration: Generalized Pentagonals In Generalized-Octagonal-Hexagrams</a>
%H Andreas Enge, William Hart and Fredrik Johansson, <a href="http://arxiv.org/abs/1608.06810">Short addition sequences for theta functions</a>, arXiv:1608.06810 [math.NT], 2016.
%H Leonhard Euler, <a href="https://scholarlycommons.pacific.edu/euler-works/175/">Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs</a>, Opera Omnia, Series I, Vol. 2 (1751), pp. 241-253.
%H Leonhard Euler, <a href="https://arxiv.org/abs/math/0505373">On the remarkable properties of the pentagonal numbers</a>, arXiv:math/0505373 [math.HO], 2005.
%H Leonhard Euler, <a href="http://math.dartmouth.edu/~euler/pages/E542.html">De mirabilibus proprietatibus numerorum pentagonalium</a>, par. 2
%H Leonhard Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">Observatio de summis divisorum</a> p. 8.
%H Leonhard Euler, <a href="https://arxiv.org/abs/math/0411587">An observation on the sums of divisors</a>, p. 8, arXiv:math/0411587 [math.HO], 2004.
%H Alex Fink, Richard K. Guy and Mark Krusemeyer, <a href="https://cdm.ucalgary.ca/article/view/61940/46659">Partitions with parts occurring at most thrice</a>, Contrib. Discr. Math., Vol. 3, No. 2 (2008), pp. 76-114.
%H Silvia Heubach and Toufik Mansour, <a href="https://arxiv.org/abs/math/0310197">Counting rises, levels and drops in compositions</a>, arXiv:math/0310197 [math.CO], 2003.
%H Alfred Hoehn, <a href="/A001318/a001318.pdf">Illustration of initial terms</a>.
%H Barbara H. Margolius, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/MARGOLIUS/inversions.html">Permutations with inversions</a>, J. Integ. Seq., Vol. 4 (2001), Article 01.2.4.
%H Johannes W. Meijer, Euler's Ship on the Pentagonal Sea, <a href="/A001318/a001318_1.pdf">pdf</a> and <a href="/A001318/a001318.jpg">jpg</a>.
%H Johannes W. Meijer and Manuel Nepveu, <a href="http://ucbconocimiento.cba.ucb.edu.bo/index.php/RAN/article/download/485/427">Euler's ship on the Pentagonal Sea</a>, Acta Nova, Vol. 4, No. 1 (December 2008), pp. 176-187.
%H Mircea Merca, <a href="https://www.researchgate.net/publication/312324402">The Lambert series factorization theorem</a>, The Ramanujan Journal, January 2017; DOI: 10.1007/s11139-016-9856-3.
%H Mircea Merca and Maxie D. Schmidt, <a href="https://arxiv.org/abs/1706.02359">New Factor Pairs for Factorizations of Lambert Series Generating Functions</a>, arXiv:1706.02359 [math.CO], 2017. See Remark 2.2.
%H Ivan Niven, <a href="http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/IvanNiven.pdf">Formal power series</a>, Amer. Math. Monthly, Vol. 76, No. 8 (1969), pp. 871-889.
%H Vladimir Pletser, <a href="http://arxiv.org/abs/1409.7969">Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers</a>, arXiv:1409.7969 [math.NT], 2014.
%H Vladimir Pletser, <a href="http://arxiv.org/abs/1409.7972">Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials</a>, arXiv preprint arXiv:1409.7972 [math.NT], 2014.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H S. Realis, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN598948236_0004&DMDID=DMDLOG_0076&IDDOC=630831">Question 271</a>, Nouv. Corresp. Math., 4 (1878) 27-29.
%H Steven J. Schlicker, <a href="http://www.jstor.org/stable/10.4169/math.mag.84.5.339">Numbers Simultaneously Polygonal and Centered Polygonal</a>, Mathematics Magazine, Vol. 84, No. 5 (December 2011), pp. 339-350.
%H André Weil, <a href="http://dx.doi.org/10.5169/seals-46896">Two lectures on number theory, past and present</a>, L'Enseign. Math., Vol. XX (1974), pp. 87-110; Oeuvres III, pp. 279-302.
%H M. Wohlgemuth, <a href="http://matheplanet.com/default3.html?article=277">Pentagon, Kartenhaus und Summenzerlegung</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal numbers</a>, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>.
%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Pentagonal number theorem">Pentagonal number theorem</a>.
%H Keke Zhang, <a href="https://arxiv.org/abs/2011.09593">Generalized Catalan numbers</a>, arXiv:2011.09593 [math.CO], 2020.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F Euler: Product_{n>=1} (1 - x^n) = Sum_{n=-oo..oo} (-1)^n*x^(n*(3*n - 1)/2).
%F A080995(a(n)) = 1: complement of A090864; A000009(a(n)) = A051044(n). - _Reinhard Zumkeller_, Apr 22 2006
%F Euler transform of length-3 sequence [2, 2, -1]. - _Michael Somos_, Mar 24 2011
%F a(-1 - n) = a(n) for all n in Z. a(2*n) = A005449(n). a(2*n - 1) = A000326(n). - _Michael Somos_, Mar 24 2011. [The extension of the recurrence to negative indices satisfies the signature (1,2,-2,-1,1), but not the definition of the sequence m*(3*m -1)/2, because there is no m such that a(-1) = 0. - _Klaus Purath_, Jul 07 2021]
%F a(n) = 3 + 2*a(n-2) - a(n-4). - _Ant King_, Aug 23 2011
%F Product_{k>0} (1 - x^k) = Sum_{k>=0} (-1)^k * x^a(k). - _Michael Somos_, Mar 24 2011
%F G.f.: x*(1 + x + x^2)/((1 + x)^2*(1 - x)^3).
%F a(n) = n*(n + 1)/6 when n runs through numbers == 0 or 2 mod 3. - _Barry E. Williams_
%F a(n) = A008805(n-1) + A008805(n-2) + A008805(n-3), n > 2. - _Ralf Stephan_, Apr 26 2003
%F Sequence consists of the pentagonal numbers (A000326), followed by A000326(n) + n and then the next pentagonal number. - _Jon Perry_, Sep 11 2003
%F a(n) = (6*n^2 + 6*n + 1)/16 - (2*n + 1)*(-1)^n/16; a(n) = A034828(n+1) - A034828(n). - _Paul Barry_, May 13 2005
%F a(n) = Sum_{k=1..floor((n+1)/2)} (n - k + 1). - _Paul Barry_, Sep 07 2005
%F a(n) = A000217(n) - A000217(floor(n/2)). - _Pierre CAMI_, Dec 09 2007
%F If n even a(n) = a(n-1) + n/2 and if n odd a(n) = a(n-1) + n, n >= 2. - _Pierre CAMI_, Dec 09 2007
%F a(n)-a(n-1) = A026741(n) and it follows that the difference between consecutive terms is equal to n if n is odd and to n/2 if n is even. Hence this is a self-generating sequence that can be simply constructed from knowledge of the first term alone. - _Ant King_, Sep 26 2011
%F a(n) = (1/2)*ceiling(n/2)*ceiling((3*n + 1)/2). - _Mircea Merca_, Jul 13 2012
%F a(n) = (A008794(n+1) + A000217(n))/2 = A002378(n) - A085787(n). - _Omar E. Pol_, Jan 12 2013
%F a(n) = floor((n + 1)/2)*((n + 1) - (1/2)*floor((n + 1)/2) - 1/2). - _Wesley Ivan Hurt_, Jan 26 2013
%F From _Oskar Wieland_, Apr 10 2013: (Start)
%F a(n) = a(n+1) - A026741(n),
%F a(n) = a(n+2) - A001651(n),
%F a(n) = a(n+3) - A184418(n),
%F a(n) = a(n+4) - A007310(n),
%F a(n) = a(n+6) - A001651(n)*3 = a(n+6) - A016051(n),
%F a(n) = a(n+8) - A007310(n)*2 = a(n+8) - A091999(n),
%F a(n) = a(n+10)- A001651(n)*5 = a(n+10)- A072703(n),
%F a(n) = a(n+12)- A007310(n)*3,
%F a(n) = a(n+14)- A001651(n)*7. (End)
%F a(n) = (A007310(n+1)^2 - 1)/24. - _Richard R. Forberg_, May 27 2013; corrected by _Zak Seidov_, Mar 14 2015; further corrected by _Jianing Song_, Oct 24 2018
%F a(n) = Sum_{i = ceiling((n+1)/2)..n} i. - _Wesley Ivan Hurt_, Jun 08 2013
%F G.f.: x*G(0), where G(k) = 1 + x*(3*k + 4)/(3*k + 2 - x*(3*k + 2)*(3*k^2 + 11*k + 10)/(x*(3*k^2 + 11*k + 10) + (k + 1)*(3*k + 4)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 16 2013
%F Sum_{n>=1} 1/a(n) = 6 - 2*Pi/sqrt(3). - _Vaclav Kotesovec_, Oct 05 2016
%F a(n) = Sum_{i=1..n} numerator(i/2) = Sum_{i=1..n} denominator(2/i). - _Wesley Ivan Hurt_, Feb 26 2017
%F a(n) = A000292(A001651(n))/A001651(n), for n>0. - _Ivan N. Ianakiev_, May 08 2018
%F a(n) = ((-5 + (-1)^n - 6n)*(-1 + (-1)^n - 6n))/96. - _José de Jesús Camacho Medina_, Jun 12 2018
%F a(n) = Sum_{k=1..n} k/gcd(k,2). - _Pedro Caceres_, Apr 23 2019
%F Quadrisection. For r = 0,1,2,3: a(r + 4*k) = 6*k^2 + sqrt(24*a(r) + 1)*k + a(r), for k >= 1, with inputs (k = 0) {0,1,2,5}. These are the sequences A049453(k), A033570(k), A033568(k+1), A049452(k+1), for k >= 0, respectively. - _Wolfdieter Lang_, Feb 12 2021
%F a(n) = a(n-4) + sqrt(24*a(n-2) + 1), n >= 4. - _Klaus Purath_, Jul 07 2021
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 6*(log(3)-1). - _Amiram Eldar_, Feb 28 2022
%F a(n) = A002620(n) + A008805(n-1). _Gary W. Adamson_, Dec 10 2022
%e G.f. = x + 2*x^2 + 5*x^3 + 7*x^4 + 12*x^5 + 15*x^6 + 22*x^7 + 26*x^8 + 35*x^9 + ...
%p A001318 := -(1+z+z**2)/(z+1)**2/(z-1)**3; # _Simon Plouffe_ in his 1992 dissertation; gives sequence without initial zero
%p A001318 := proc(n) (6*n^2+6*n+1)/16-(2*n+1)*(-1)^n/16 ; end proc: # _R. J. Mathar_, Mar 27 2011
%t Table[n*(n+1)/6, {n, Select[Range[0, 100], Mod[#, 3] != 1 &]}]
%t Select[Accumulate[Range[0,200]]/3,IntegerQ] (* _Harvey P. Dale_, Oct 12 2014 *)
%t CoefficientList[Series[x (1 + x + x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 70}], x] (* _Vincenzo Librandi_, Nov 04 2014 *)
%t LinearRecurrence[{1,2,-2,-1,1},{0,1,2,5,7},70] (* _Harvey P. Dale_, Jun 05 2017 *)
%t a[ n_] := With[{m = Quotient[n + 1, 2]}, m (3 m + (-1)^n) / 2]; (* _Michael Somos_, Jun 02 2018 *)
%o (PARI) {a(n) = (3*n^2 + 2*n + (n%2) * (2*n + 1)) / 8}; /* _Michael Somos_, Mar 24 2011 */
%o (PARI) {a(n) = if( n<0, n = -1-n); polcoeff( x * (1 - x^3) / ((1 - x) * (1-x^2))^2 + x * O(x^n), n)}; /* _Michael Somos_, Mar 24 2011 */
%o (PARI) {a(n) = my(m = (n+1) \ 2); m * (3*m + (-1)^n) / 2}; /* _Michael Somos_, Jun 02 2018 */
%o (Sage)
%o @CachedFunction
%o def A001318(n):
%o if n == 0 : return 0
%o inc = n//2 if is_even(n) else n
%o return inc + A001318(n-1)
%o [A001318(n) for n in (0..59)] # _Peter Luschny_, Oct 13 2012
%o (Magma) [(6*n^2 + 6*n + 1 - (2*n + 1)*(-1)^n)/16 : n in [0..50]]; // _Wesley Ivan Hurt_, Nov 03 2014
%o (Magma) [(3*n^2 + 2*n + (n mod 2) * (2*n + 1)) div 8: n in [0..70]]; // _Vincenzo Librandi_, Nov 04 2014
%o (Haskell)
%o a001318 n = a001318_list !! n
%o a001318_list = scanl1 (+) a026741_list -- _Reinhard Zumkeller_, Nov 15 2015
%o (GAP) a:=[0,1,2,5];; for n in [5..60] do a[n]:=2*a[n-2]-a[n-4]+3; od; a; # _Muniru A Asiru_, Aug 16 2018
%o (Python)
%o def a(n):
%o p = n % 2
%o return (n + p)*(3*n + 2 - p) >> 3
%o print([a(n) for n in range(60)]) # _Peter Luschny_, Jul 15 2022
%Y Cf. A080995 (characteristic function), A026741 (first differences), A034828 (partial sums), A165211 (mod 2).
%Y Cf. A000326 (pentagonal numbers), A005449 (second pentagonal numbers), A000217 (triangular numbers).
%Y Indices of nonzero terms of A010815, i.e., the (zero-based) indices of 1-bits of the infinite binary word to which the terms of A068052 converge.
%Y Union of A036498 and A036499.
%Y Cf. A153384, A168258, A101688, A174739, A175005.
%Y Cf. A074378, A057569, A057570, A007310.
%Y Sequences of generalized k-gonal numbers: this sequence (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (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).
%Y Column 1 of A195152.
%Y Squares in APs: A221671, A221672.
%Y Cf. A054440, A260664, A260672.
%Y Quadrisection: A049453(k), A033570(k), A033568(k+1), A049452(k+1), k >= 0.
%Y Cf. A002620.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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