%I #7 Feb 10 2021 20:40:00

%S 1,7,22,42,63,99,160,220,265,337,457,577,672,792,978,1178,1319,1469,

%T 1739,2039,2255,2507,2882,3242,3513,3819,4269,4769,5159,5555,6181,

%U 6841,7246,7666,8401,9181,9763,10363,11188,12108,12828,13434,14394,15534,16359,17211,18477,19677

%N Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_6)^2 <= n.

%C Partial sums of A045848.

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

%F G.f.: (1 + theta_3(x))^6 / (64 * (1 - x)).

%F a(n^2) = A055405(n).

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

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

%p end:

%p a:= proc(n) option remember; b(n, 6)+`if`(n>0, a(n-1), 0) end:

%p seq(a(n), n=0..47); # _Alois P. Heinz_, Feb 10 2021

%t nmax = 47; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^6/(64 (1 - x)), {x, 0, nmax}], x]

%Y Cf. A000122, A000606, A003059, A045848, A055405, A055412, A175361, A224212, A224213, A302862, A341400, A341402.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Feb 10 2021