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%I #14 Dec 21 2023 15:47:09
%S 0,0,1,1,5,11,32,44,82,120,207,277,405,541,768,966,1272,1592,2087,
%T 2489,3103,3719,4588,5348,6386,7522,8891,10175,11909,13623,15818,
%U 17742,20278,22720,25923,28917,32361,36031,40368,44488,49400,54358,60377,65835,72341
%N Number of solutions to x^2 + y^2 + z^2 + w^2 <= n^2, where x, y, z, w are positive odd integers.
%H Robert Israel, <a href="/A349611/b349611.txt">Table of n, a(n) for n = 0..2500</a>
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F a(n) = [x^(n^2)] theta_2(x^4)^4 / (16 * (1 - x)).
%e a(4) = 5 since there are solutions (1,1,1,1), (3,1,1,1), (1,3,1,1), (1,1,3,1), (1,1,1,3).
%p N:= 100: # for a(0) .. a(N)
%p F:= add(x^(k^2),k = 1 ... N,2):
%p F:= expand(F^4):
%p L:= ListTools:-PartialSums([seq](coeff(F,x,n),n=0..N^2)):
%p L[[seq(n^2+1,n=0..N)]]; # _Robert Israel_, Dec 21 2023
%t Table[SeriesCoefficient[EllipticTheta[2, 0, x^4]^4/(16 (1 - x)), {x, 0, n^2}], {n, 0, 44}]
%Y Cf. A055403, A055410, A341423, A349609, A349610.
%K nonn
%O 0,5
%A _Ilya Gutkovskiy_, Nov 23 2021
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