A097057 -id:A097057 - cp's OEIS frontend

A320124 Number of integer solutions to a^2 + b^2 + 2c^2 + 3d^2 = n.

1, 4, 6, 10, 20, 20, 24, 40, 22, 28, 56, 20, 50, 80, 28, 80, 84, 32, 78, 80, 68, 100, 120, 80, 88, 124, 56, 82, 136, 100, 140, 200, 86, 80, 192, 72, 140, 240, 120, 200, 248, 80, 112, 176, 100, 260, 224, 160, 210, 172, 186, 128, 272, 180, 240, 400, 124, 200, 280, 116

Offset: 0

Seiichi Manyama, Oct 06 2018

nonn

a(n) > 0 for n >= 0.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A097057, A320125, A320126, A320127, A320128.

G.f.: theta_3(q)^2 * theta_3(q^2) * theta_3(q^3).

A320125 Number of integer solutions to a^2 + b^2 + 2c^2 + 4d^2 = n.

Original entry on oeis.org

1, 4, 6, 8, 14, 16, 20, 32, 30, 28, 40, 40, 36, 48, 48, 32, 62, 72, 42, 72, 72, 64, 100, 96, 68, 84, 120, 80, 112, 112, 48, 128, 126, 80, 108, 128, 98, 144, 180, 96, 136, 168, 160, 168, 180, 112, 144, 192, 132, 228, 126, 144, 216, 208, 200, 160, 240, 144, 280, 232

Offset: 0

Seiichi Manyama, Oct 06 2018

nonn

a(n) > 0 for n >= 0.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A097057, A320124, A320126, A320127, A320128.

G.f.: theta_3(q)^2 * theta_3(q^2) * theta_3(q^4).

A320126 Number of integer solutions to a^2 + b^2 + 2c^2 + 5d^2 = n.

Original entry on oeis.org

1, 4, 6, 8, 12, 10, 16, 28, 22, 36, 40, 24, 56, 36, 24, 64, 28, 64, 78, 24, 58, 48, 68, 92, 80, 92, 72, 112, 92, 48, 104, 96, 118, 176, 64, 48, 124, 84, 148, 160, 104, 120, 176, 120, 72, 146, 88, 204, 216, 124, 126, 96, 180, 148, 224, 188, 120, 304, 216, 120, 224, 96

Offset: 0

Seiichi Manyama, Oct 06 2018

nonn

a(n) > 0 for n >= 0.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A097057, A320124, A320125, A320127, A320128.

G.f.: theta_3(q)^2 * theta_3(q^2) * theta_3(q^5).

A320127 Number of integer solutions to a^2 + b^2 + 2c^2 + 6d^2 = n.

Original entry on oeis.org

1, 4, 6, 8, 12, 8, 10, 24, 18, 28, 48, 24, 40, 56, 12, 40, 60, 32, 78, 72, 24, 48, 48, 48, 70, 84, 84, 80, 120, 56, 32, 120, 66, 120, 192, 48, 84, 152, 36, 112, 144, 80, 180, 168, 120, 104, 96, 96, 136, 172, 126, 160, 168, 104, 82, 192, 84, 144, 336, 120, 200, 248, 60, 168

Offset: 0

Seiichi Manyama, Oct 06 2018

nonn

a(n) > 0 for n >= 0.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A097057, A320124, A320125, A320126, A320128.

G.f.: theta_3(q)^2 * theta_3(q^2) * theta_3(q^6).

A320128 Number of integer solutions to a^2 + b^2 + 2c^2 + 7d^2 = n.

Original entry on oeis.org

1, 4, 6, 8, 12, 8, 8, 18, 14, 24, 40, 32, 40, 40, 32, 28, 36, 64, 46, 72, 72, 16, 56, 40, 40, 88, 72, 80, 82, 64, 44, 64, 86, 96, 128, 144, 64, 64, 136, 44, 136, 160, 80, 144, 128, 120, 92, 128, 104, 164, 158, 96, 168, 80, 144, 160, 168, 108, 104, 232, 108, 168, 192, 82

Offset: 0

Seiichi Manyama, Oct 06 2018

nonn

a(n) > 0 for n >= 0.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A097057, A320124, A320125, A320126, A320127.

G.f.: theta_3(q)^2 * theta_3(q^2) * theta_3(q^7).

A236923 Number of integer solutions to a^2 + b^2 + c^2 + 4*d^2 = n.

Original entry on oeis.org

1, 6, 12, 8, 8, 36, 48, 16, 24, 78, 72, 24, 32, 84, 96, 48, 24, 108, 156, 40, 48, 192, 144, 48, 96, 186, 168, 80, 64, 180, 288, 64, 24, 288, 216, 96, 104, 228, 240, 112, 144, 252, 384, 88, 96, 468, 288, 96, 96, 342, 372, 144, 112, 324, 480, 144, 192, 480, 360, 120, 192, 372, 384, 208, 24, 504, 576, 136, 144, 576, 576, 144, 312, 444, 456, 248, 160, 576, 672

Offset: 0

N. J. A. Sloane, Feb 14 2014

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Olivia X. M. Yao and Ernest X. W. Xia, Combinatorial proofs of five formulas of Liouville, Discrete Math. 318 (2014), 1--9. MR3141622.

Cf. A097057, A236924.

For number of solutions to a^2+b^2+c^2+k*d^2=n for k=1,2,3,4,5,6,7,8,12 see A000118, A236928, A236926, A236923, A236930, A236931, A236932, A236927, A236933.

with(numtheory);
s:=n-> if whattype(n) = integer then sigma(n) else 0; fi;
f:=proc(n) global s;
  if (n mod 4) = 0 then 8*s(n/4)-32*s(n/16)
elif (n mod 4) = 2 then 12*s(n/2)
elif (n mod 4) = 3 then 2*s(n)
else 6*s(n);
fi; end;
[seq(f(n),n=1..100)];
# a(0)=1 must be added separately

EllipticTheta[3, 0, q]^3*EllipticTheta[3, 0, q^4] + O[q]^80 // CoefficientList[#, q]& (* Jean-François Alcover, Mar 04 2023, after Ilya Gutkovskiy *)

See Maple code.

G.f.: theta_3(q)^3*theta_3(q^4), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

A320130 Number of integer solutions to a^2 + b^2 + 2c^2 + 8d^2 = n.

Original entry on oeis.org

1, 4, 6, 8, 12, 8, 8, 16, 8, 20, 36, 24, 48, 40, 16, 48, 24, 40, 78, 40, 72, 64, 24, 48, 32, 60, 84, 80, 96, 40, 48, 64, 24, 96, 108, 96, 156, 72, 40, 112, 48, 104, 192, 88, 144, 168, 48, 96, 96, 100, 186, 144, 168, 136, 80, 144, 64, 160, 180, 120, 288, 104, 64, 208

Offset: 0

Seiichi Manyama, Oct 06 2018

nonn

a(n) > 0 for n >= 0.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A097057, A320124, A320125, A320126, A320127, A320128, A320131, A320132, A320133, A320134, A320135, A320136.

G.f.: theta_3(q)^2 * theta_3(q^2) * theta_3(q^8).

A320131 Number of integer solutions to a^2 + b^2 + 2c^2 + 9d^2 = n.

Original entry on oeis.org

1, 4, 6, 8, 12, 8, 8, 16, 6, 14, 32, 20, 40, 48, 16, 32, 44, 28, 54, 72, 40, 64, 72, 16, 40, 52, 56, 92, 96, 56, 32, 80, 38, 48, 104, 64, 102, 144, 52, 32, 112, 44, 128, 168, 68, 136, 128, 96, 104, 132, 62, 144, 176, 104, 172, 96, 112, 112, 192, 116, 96, 240, 96, 176

Offset: 0

Seiichi Manyama, Oct 06 2018

nonn

a(n) > 0 for n >= 0.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A097057, A320124, A320125, A320126, A320127, A320128, A320130, A320132, A320133, A320134, A320135, A320136.

G.f.: theta_3(q)^2 * theta_3(q^2) * theta_3(q^9).

A320132 Number of integer solutions to a^2 + b^2 + 2c^2 + 10d^2 = n.

Original entry on oeis.org

1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 26, 16, 36, 40, 24, 32, 28, 48, 42, 48, 72, 32, 72, 64, 8, 60, 48, 64, 108, 56, 48, 64, 54, 64, 64, 72, 84, 104, 120, 32, 26, 88, 72, 120, 144, 88, 88, 144, 84, 92, 126, 64, 216, 152, 80, 160, 24, 128, 180, 144, 156, 104, 192, 128, 92

Offset: 0

Seiichi Manyama, Oct 06 2018

nonn

a(n) > 0 for n >= 0.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A097057, A320124, A320125, A320126, A320127, A320128, A320130, A320131, A320133, A320134, A320135, A320136.

G.f.: theta_3(q)^2 * theta_3(q^2) * theta_3(q^10).

A320133 Number of integer solutions to a^2 + b^2 + 2c^2 + 11d^2 = n.

Original entry on oeis.org

1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 24, 10, 32, 36, 16, 40, 28, 32, 62, 36, 48, 64, 40, 64, 56, 28, 56, 56, 80, 68, 48, 80, 38, 80, 80, 32, 92, 88, 88, 112, 40, 16, 112, 52, 90, 144, 44, 120, 128, 100, 78, 96, 116, 144, 160, 112, 128, 80, 136, 104, 144, 132, 92, 240

Offset: 0

Seiichi Manyama, Oct 06 2018

nonn

a(n) > 0 for n >= 0.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A097057, A320124, A320125, A320126, A320127, A320128, A320130, A320131, A320132, A320134, A320135, A320136.

G.f.: theta_3(q)^2 * theta_3(q^2) * theta_3(q^11).

cp's OEIS Frontend

A320124 Number of integer solutions to a^2 + b^2 + 2*c^2 + 3*d^2 = n.

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A320125 Number of integer solutions to a^2 + b^2 + 2*c^2 + 4*d^2 = n.

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A320126 Number of integer solutions to a^2 + b^2 + 2*c^2 + 5*d^2 = n.

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A320127 Number of integer solutions to a^2 + b^2 + 2*c^2 + 6*d^2 = n.

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A320128 Number of integer solutions to a^2 + b^2 + 2*c^2 + 7*d^2 = n.

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A236923 Number of integer solutions to a^2 + b^2 + c^2 + 4*d^2 = n.

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A320130 Number of integer solutions to a^2 + b^2 + 2*c^2 + 8*d^2 = n.

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A320131 Number of integer solutions to a^2 + b^2 + 2*c^2 + 9*d^2 = n.

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A320132 Number of integer solutions to a^2 + b^2 + 2*c^2 + 10*d^2 = n.

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A320133 Number of integer solutions to a^2 + b^2 + 2*c^2 + 11*d^2 = n.

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A320124 Number of integer solutions to a^2 + b^2 + 2c^2 + 3d^2 = n.

A320125 Number of integer solutions to a^2 + b^2 + 2c^2 + 4d^2 = n.

A320126 Number of integer solutions to a^2 + b^2 + 2c^2 + 5d^2 = n.

A320127 Number of integer solutions to a^2 + b^2 + 2c^2 + 6d^2 = n.

A320128 Number of integer solutions to a^2 + b^2 + 2c^2 + 7d^2 = n.

A320130 Number of integer solutions to a^2 + b^2 + 2c^2 + 8d^2 = n.

A320131 Number of integer solutions to a^2 + b^2 + 2c^2 + 9d^2 = n.

A320132 Number of integer solutions to a^2 + b^2 + 2c^2 + 10d^2 = n.

A320133 Number of integer solutions to a^2 + b^2 + 2c^2 + 11d^2 = n.