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A341396 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n.

%I A341396 #12 Jun 21 2024 02:46:13
%S A341396 1,15,99,379,953,1793,3081,5449,8893,12435,16859,24419,33659,42115,
%T A341396 53203,69779,88273,106081,125821,153541,187981,217437,248741,298469,
%U A341396 351277,394691,446939,515259,589307,657683,728803,828259,939223,1029159,1124023,1260103
%N A341396 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n.
%C A341396 Partial sums of A008451.
%H A341396 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A341396 G.f.: theta_3(x)^7 / (1 - x).
%F A341396 a(n^2) = A055413(n).
%p A341396 b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
%p A341396       b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
%p A341396     end:
%p A341396 a:= proc(n) option remember; b(n, 7)+`if`(n>0, a(n-1), 0) end:
%p A341396 seq(a(n), n=0..35);  # _Alois P. Heinz_, Feb 10 2021
%t A341396 nmax = 35; CoefficientList[Series[EllipticTheta[3, 0, x]^7/(1 - x), {x, 0, nmax}], x]
%t A341396 Table[SquaresR[7, n], {n, 0, 35}] // Accumulate
%o A341396 (PARI) my(q='q+O('q^(55))); Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^7/(1-q)) \\ _Joerg Arndt_, Jun 21 2024
%Y A341396 Cf. A000122, A001650, A008451, A046895, A055406, A055413, A057655, A117609, A122510, A175360, A175361, A302860, A341397, A341398, A341399.
%K A341396 nonn
%O A341396 0,2
%A A341396 _Ilya Gutkovskiy_, Feb 10 2021