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

0,5

Links

Robert Israel, Table of n, a(n) for n = 0..2500

Index entries for sequences related to sums of squares

Formula

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

Example

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).

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}]

Crossrefs

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

Sequence in context: A236428 A106908 A107442 * A323867 A280540 A077917

Adjacent sequences: A349608 A349609 A349610 * A349612 A349613 A349614

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 23 2021

Status

approved