A349609 -id:A349609 - cp's OEIS frontend

A349610 Number of solutions to x^2 + y^2 + z^2 <= n^2, where x, y, z are positive odd integers.

0, 0, 1, 1, 4, 7, 17, 20, 35, 45, 69, 84, 114, 136, 184, 217, 272, 314, 389, 443, 528, 597, 702, 784, 901, 1018, 1166, 1268, 1442, 1589, 1791, 1926, 2157, 2332, 2584, 2800, 3058, 3293, 3596, 3872, 4194, 4485, 4878, 5184, 5590, 5950, 6388, 6761, 7232

Offset: 0

Ilya Gutkovskiy, Nov 23 2021

nonn

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

David A. Corneth, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of squares

Cf. A000604, A000605, A008437, A053596, A085121, A253663, A349609, A349611.

Table[SeriesCoefficient[EllipticTheta[2, 0, x^4]^3/(8 (1 - x)), {x, 0, n^2}], {n, 0, 48}]

a(n) = [x^(n^2)] theta_2(x^4)^3 / (8 * (1 - x)).
a(n) = Sum_{k=0..n^2} A008437(k).
a(n) = A053596(n) / 8.

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

Original entry on oeis.org

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

Ilya Gutkovskiy, Nov 23 2021

nonn

			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).

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

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

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

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

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

A372511 Number of solutions to x^2 + y^2 <= n, where x, y are positive odd integers.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17

Offset: 0

Ilya Gutkovskiy, May 04 2024

nonn

Index entries for sequences related to sums of squares

Cf. A008442, A057655, A224212, A237526, A290081, A349609, A372512.

nmax = 85; CoefficientList[Series[EllipticTheta[2, 0, x^4]^2/(4 (1 - x)), {x, 0, nmax}], x]

cp's OEIS Frontend

A349610 Number of solutions to x^2 + y^2 + z^2 <= n^2, where x, y, z are positive odd integers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A372511 Number of solutions to x^2 + y^2 <= n, where x, y are positive odd integers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica