A349611 Number of solutions to x^2 + y^2 + z^2 + w^2 <= n^2, where x, y, z, w are positive odd integers.

0, 0, 1, 1, 5, 11, 32, 44, 82, 120, 207, 277, 405, 541, 768, 966, 1272, 1592, 2087, 2489, 3103, 3719, 4588, 5348, 6386, 7522, 8891, 10175, 11909, 13623, 15818, 17742, 20278, 22720, 25923, 28917, 32361, 36031, 40368, 44488, 49400, 54358, 60377, 65835, 72341

Offset: 0

Author

Ilya Gutkovskiy, Nov 23 2021

Keywords

• nonn

Examples

			a(4) = 5 since there are solutions (1,1,1,1), (3,1,1,1), (1,3,1,1), (1,1,3,1), (1,1,1,3).
		

Links

• Robert Israel, Table of n, a(n) for n = 0..2500
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A055403, A055410, A341423, A349609, A349610.

Programs

Maple

N:= 100: # for a(0) .. a(N)
F:= add(x^(k^2),k = 1 ... N,2):
F:= expand(F^4):
L:= ListTools:-PartialSums([seq](coeff(F,x,n),n=0..N^2)):
L[[seq(n^2+1,n=0..N)]]; # Robert Israel, Dec 21 2023

Mathematica

Table[SeriesCoefficient[EllipticTheta[2, 0, x^4]^4/(16 (1 - x)), {x, 0, n^2}], {n, 0, 44}]

Formula

a(n) = [x^(n^2)] theta_2(x^4)^4 / (16 * (1 - x)).