cp's OEIS Frontend

A100255 Squares of pentagonal numbers: a(n) = (1/4)*n^2*(3*n-1)^2.

%I A100255 #29 Apr 04 2025 04:18:07
%S A100255 0,1,25,144,484,1225,2601,4900,8464,13689,21025,30976,44100,61009,
%T A100255 82369,108900,141376,180625,227529,283024,348100,423801,511225,611524,
%U A100255 725904,855625,1002001,1166400,1350244,1555009,1782225,2033476
%N A100255 Squares of pentagonal numbers: a(n) = (1/4)*n^2*(3*n-1)^2.
%C A100255 More generally, the ordinary generating function for the squares of k-gonal numbers is x*(1 + (k^2 - 5)*x + (4*k^2 - 18*k + 19)*x^2 + (k - 3)^2*x^3)/(1 - x)^5. - _Ilya Gutkovskiy_, Apr 13 2016
%H A100255 Michael De Vlieger, <a href="/A100255/b100255.txt">Table of n, a(n) for n = 0..10000</a>
%H A100255 Leonhard Euler, <a href="https://scholarlycommons.pacific.edu/euler-works/542/">De mirabilibus proprietatibus numerorum pentagonalium</a>, The Euler Archive, 1783, par. 29.
%H A100255 Leonhard Euler, <a href="http://arXiv.org/abs/math.HO/0505373">On the remarkable properties of the pentagonal numbers</a>, arXiv:math/0505373 [math.HO], 2005.
%H A100255 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A100255 a(n) = A000326(n)^2.
%F A100255 G.f.: x*(1+20*x+29*x^2+4*x^3)/(1-x)^5. - _Colin Barker_, Feb 14 2012
%F A100255 From _Ilya Gutkovskiy_, Apr 13 2016: (Start)
%F A100255 E.g.f.: x*(4 + 46*x + 48*x^2 + 9*x^3)*exp(x)/4.
%F A100255 a(n) = 5*a(n-1) - 10*(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
%F A100255 Sum_{n>=1} 1/a(n) = 2*Pi^2/3 + 4*sqrt(3)*Pi - 36*log(3) + 4*psi_1(2/3), where psi_1 is the trigamma function. - _Amiram Eldar_, Apr 04 2025
%t A100255 LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 25, 144, 484}, 32] (* _Ilya Gutkovskiy_, Apr 13 2016 *)
%t A100255 Table[(1/4) n^2 (3 n - 1)^2, {n, 0, 31}] (* _Michael De Vlieger_, Apr 13 2016 *)
%o A100255 (PARI) a(n) = (1/4)*n^2*(3*n-1)^2 \\ _Altug Alkan_, Apr 13 2016
%Y A100255 Cf. A000326, A100256.
%Y A100255 Cf. similar sequences of the squares of k-gonal numbers:  A000537 (k = 3), A000583 (k = 4), this sequence (k = 5).
%K A100255 nonn,easy
%O A100255 0,3
%A A100255 _Ralf Stephan_, Nov 13 2004