A276285 - OEIS

%I #15 May 27 2017 11:08:32

%S 1,26,312,2288,11466,41808,116688,265408,535704,1031914,1899664,

%T 3214224,5043376,7801744,12066912,17689152,24443978,34039200,48210760,

%U 64966096,83323344,109157152,145532816,185245632,227110416,284788010,363737712

%N Number of ways of writing n as a sum of 13 squares.

%C More generally, the ordinary generating function for the number of ways of writing n as a sum of k squares is theta_3(0, q)^k = 1 + 2*k*q + 2*(k - 1)*k*q^2 + (4/3)*(k - 2)*(k - 1)*k*q^3 + (2/3)*((k - 3)*(k - 2)*(k - 1) + 3)*k*q^4 + (4/15) *(k - 1)*k*(k^3 - 9*k^2 + 26*k - 9)*q^5 + ..., where theta is the Jacobi theta functions.

%H Seiichi Manyama, <a href="/A276285/b276285.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SumofSquaresFunction.html">Sum of Squares Function</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F G.f.: theta_3(0,q)^13, where theta_3(x,q) is the third Jacobi theta function.

%F a(n) = (26/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, May 27 2017

%t Table[SquaresR[13, n], {n, 0, 26}]

%Y 13th column of A286815. - _Seiichi Manyama_, May 27 2017

%Y Row d=13 of A122141.

%Y Cf. Number of ways of writing n as a sum of k squares: A004018 (k = 2), A005875 (k = 3), A000118 (k = 4), A000132 (k = 5), A000141 (k = 6), A008451 (k = 7), A000143 (k = 8), A008452 (k = 9), A000144 (k = 10), A008453 (k = 11), A000145 (k = 12), this sequence (k = 13), A000152 (k = 16).

%K nonn,easy

%O 0,2

%A _Ilya Gutkovskiy_, Aug 27 2016