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A341398 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.

%I A341398 #8 Feb 10 2021 19:29:17
%S A341398 1,19,163,835,2869,7189,14581,27253,49861,84663,129303,190071,284055,
%T A341398 409335,550455,732855,995241,1312617,1656153,2077497,2634777,3300057,
%U A341398 4003641,4804281,5872665,7129227,8363307,9784491,11635755,13670475,15727755,18066315,20950491
%N A341398 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.
%C A341398 Partial sums of A008452.
%H A341398 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A341398 G.f.: theta_3(x)^9 / (1 - x).
%F A341398 a(n^2) = A055415(n).
%p A341398 b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
%p A341398       b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
%p A341398     end:
%p A341398 a:= proc(n) option remember; b(n, 9)+`if`(n>0, a(n-1), 0) end:
%p A341398 seq(a(n), n=0..32);  # _Alois P. Heinz_, Feb 10 2021
%t A341398 nmax = 32; CoefficientList[Series[EllipticTheta[3, 0, x]^9/(1 - x), {x, 0, nmax}], x]
%t A341398 Table[SquaresR[9, n], {n, 0, 32}] // Accumulate
%Y A341398 Cf. A000122, A001650, A008452, A046895, A055408, A055415, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341397, A341399.
%K A341398 nonn
%O A341398 0,2
%A A341398 _Ilya Gutkovskiy_, Feb 10 2021