%I #12 Jun 21 2024 02:46:13

%S 1,15,99,379,953,1793,3081,5449,8893,12435,16859,24419,33659,42115,

%T 53203,69779,88273,106081,125821,153541,187981,217437,248741,298469,

%U 351277,394691,446939,515259,589307,657683,728803,828259,939223,1029159,1124023,1260103

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

%C Partial sums of A008451.

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F G.f.: theta_3(x)^7 / (1 - x).

%F a(n^2) = A055413(n).

%p b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,

%p b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))

%p end:

%p a:= proc(n) option remember; b(n, 7)+`if`(n>0, a(n-1), 0) end:

%p seq(a(n), n=0..35); # _Alois P. Heinz_, Feb 10 2021

%t nmax = 35; CoefficientList[Series[EllipticTheta[3, 0, x]^7/(1 - x), {x, 0, nmax}], x]

%t Table[SquaresR[7, n], {n, 0, 35}] // Accumulate

%o (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 Cf. A000122, A001650, A008451, A046895, A055406, A055413, A057655, A117609, A122510, A175360, A175361, A302860, A341397, A341398, A341399.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Feb 10 2021