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

%I

%S 0,1,25,144,484,1225,2601,4900,8464,13689,21025,30976,44100,61009,

%T 82369,108900,141376,180625,227529,283024,348100,423801,511225,611524,

%U 725904,855625,1002001,1166400,1350244,1555009,1782225,2033476

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

%C 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 Michael De Vlieger, <a href="/A100255/b100255.txt">Table of n, a(n) for n = 0..10000</a>

%H L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E542.html">De mirabilibus proprietatibus numerorum pentagonalium</a>, par. 29

%H L. 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 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1)

%F a(n) = A000326(n)^2.

%F G.f.: x*(1+20*x+29*x^2+4*x^3)/(1-x)^5. [_Colin Barker_, Feb 14 2012]

%F From _Ilya Gutkovskiy_, Apr 13 2016: (Start)

%F E.g.f.: x*(4 + 46*x + 48*x^2 + 9*x^3)*exp(x)/4.

%F a(n) = 5*a(n-1) - 10*(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)

%t LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 25, 144, 484}, 32] (* _Ilya Gutkovskiy_, Apr 13 2016 *)

%t Table[(1/4) n^2 (3 n - 1)^2, {n, 0, 31}] (* _Michael De Vlieger_, Apr 13 2016 *)

%o (PARI) a(n) = (1/4)*n^2*(3*n-1)^2 \\ _Altug Alkan_, Apr 13 2016

%Y Cf. A000326, A100256.

%Y Cf. similar sequences of the squares of k-gonal numbers: A000537 (k = 3), A000583 (k = 4), this sequence (k = 5).

%K nonn,easy

%O 0,3

%A _Ralf Stephan_, Nov 13 2004

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Last modified January 17 22:51 EST 2019. Contains 319251 sequences. (Running on oeis4.)