A349611 -id:A349611 - 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.

A349609 Number of solutions to x^2 + y^2 <= n^2, where x, y are positive odd integers.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 8, 13, 15, 20, 22, 28, 31, 39, 43, 52, 54, 64, 69, 79, 83, 96, 102, 112, 121, 135, 140, 154, 162, 179, 185, 203, 212, 228, 238, 255, 265, 281, 296, 316, 326, 349, 359, 382, 394, 416, 429, 451, 469, 494, 508, 532, 547, 573, 587

Offset: 0

Ilya Gutkovskiy, Nov 23 2021

nonn

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

Index entries for sequences related to sums of squares

Cf. A000328, A000603, A001182, A008442, A053415, A290081, A349610, A349611.

Table[SeriesCoefficient[EllipticTheta[2, 0, x^4]^2/(4 (1 - x)), {x, 0, n^2}], {n, 0, 55}]

a(n) = [x^(n^2)] theta_2(x^4)^2 / (4 * (1 - x)).
a(n) = Sum_{k=0..n^2} A290081(k).
a(n) = A053415(n) / 4.

A372613 Number of solutions to x^2 + y^2 + z^2 + w^2 <= n, where x, y, z, w are positive odd integers.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 5, 5, 5, 5, 5, 5, 11, 11, 11, 11, 11, 11, 11, 11, 19, 19, 19, 19, 19, 19, 19, 19, 32, 32, 32, 32, 32, 32, 32, 32, 44, 44, 44, 44, 44, 44, 44, 44, 58, 58, 58, 58, 58, 58, 58, 58, 82, 82, 82, 82, 82, 82, 82, 82, 100, 100, 100, 100, 100, 100, 100, 100

Offset: 0

Ilya Gutkovskiy, May 07 2024

nonn

Index entries for sequences related to sums of squares

Cf. A008438, A046895, A224213, A349611, A372511, A372512.

nmax = 75; CoefficientList[Series[EllipticTheta[2, 0, x^4]^4/(16 (1 - x)), {x, 0, nmax}], x]

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

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A349609 Number of solutions to x^2 + y^2 <= n^2, where x, y are positive odd integers.

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A372613 Number of solutions to x^2 + y^2 + z^2 + w^2 <= n, where x, y, z, w are positive odd integers.

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