cp's OEIS Frontend

A276285 Number of ways of writing n as a sum of 13 squares.

%I A276285 #16 Feb 16 2025 08:33:36
%S A276285 1,26,312,2288,11466,41808,116688,265408,535704,1031914,1899664,
%T A276285 3214224,5043376,7801744,12066912,17689152,24443978,34039200,48210760,
%U A276285 64966096,83323344,109157152,145532816,185245632,227110416,284788010,363737712
%N A276285 Number of ways of writing n as a sum of 13 squares.
%C A276285 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 A276285 Seiichi Manyama, <a href="/A276285/b276285.txt">Table of n, a(n) for n = 0..10000</a>
%H A276285 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SumofSquaresFunction.html">Sum of Squares Function</a>
%H A276285 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A276285 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A276285 G.f.: theta_3(0,q)^13, where theta_3(x,q) is the third Jacobi theta function.
%F A276285 a(n) = (26/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, May 27 2017
%t A276285 Table[SquaresR[13, n], {n, 0, 26}]
%Y A276285 13th column of A286815. - _Seiichi Manyama_, May 27 2017
%Y A276285 Row d=13 of A122141.
%Y A276285 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 A276285 nonn,easy
%O A276285 0,2
%A A276285 _Ilya Gutkovskiy_, Aug 27 2016