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A341397 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n.

%I A341397 #15 Jun 21 2024 17:18:59
%S A341397 1,17,129,577,1713,3729,6865,12369,21697,33809,47921,69233,101041,
%T A341397 136209,174737,231185,306049,384673,469457,579217,722353,876465,
%U A341397 1025649,1220337,1481521,1733537,1979713,2306753,2697537,3087777,3482913,3959585,4558737,5155473
%N A341397 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n.
%C A341397 Partial sums of A000143.
%H A341397 Alois P. Heinz, <a href="/A341397/b341397.txt">Table of n, a(n) for n = 0..10000</a>
%H A341397 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A341397 G.f.: theta_3(x)^8 / (1 - x).
%F A341397 a(n^2) = A055414(n).
%p A341397 b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
%p A341397       b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
%p A341397     end:
%p A341397 a:= proc(n) option remember; b(n, 8)+`if`(n>0, a(n-1), 0) end:
%p A341397 seq(a(n), n=0..33);  # _Alois P. Heinz_, Feb 10 2021
%t A341397 nmax = 33; CoefficientList[Series[EllipticTheta[3, 0, x]^8/(1 - x), {x, 0, nmax}], x]
%t A341397 Table[SquaresR[8, n], {n, 0, 33}] // Accumulate
%o A341397 (Python)
%o A341397 from math import prod
%o A341397 from sympy import factorint
%o A341397 def A341397(n): return (sum((prod((p**(3*(e+1))-(1 if p&1 else 15))//(p**3-1) for p, e in factorint(m).items()) for m in range(1,n+1)))<<4)+1 # _Chai Wah Wu_, Jun 21 2024
%Y A341397 Cf. A000122, A000143, A001650, A046895, A055407, A055414, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341398, A341399.
%K A341397 nonn
%O A341397 0,2
%A A341397 _Ilya Gutkovskiy_, Feb 10 2021