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A341403 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n.

%I A341403 #6 Feb 10 2021 20:39:37
%S A341403 1,9,37,93,171,283,479,767,1076,1420,1952,2688,3444,4228,5320,6776,
%T A341403 8262,9662,11454,13918,16480,18832,21772,25644,29508,33044,37300,
%U A341403 42732,48340,53556,59632,67472,75405,82237,90189,100661,111155,120403,131099,144651,158469,170621
%N A341403 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n.
%C A341403 Partial sums of A045850.
%H A341403 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A341403 G.f.: (1 + theta_3(x))^8 / (256 * (1 - x)).
%F A341403 a(n^2) = A055407(n).
%p A341403 b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
%p A341403       b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
%p A341403     end:
%p A341403 a:= proc(n) option remember; b(n, 8)+`if`(n>0, a(n-1), 0) end:
%p A341403 seq(a(n), n=0..41);  # _Alois P. Heinz_, Feb 10 2021
%t A341403 nmax = 41; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^8/(256 (1 - x)), {x, 0, nmax}], x]
%Y A341403 Cf. A000122, A000606, A003059, A045850, A055407, A055414, A224212, A224213, A302862, A341400, A341401, A341402, A341404, A341405.
%K A341403 nonn
%O A341403 0,2
%A A341403 _Ilya Gutkovskiy_, Feb 10 2021