A236924 -id:A236924 - cp's OEIS frontend

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

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

A236928 Number of integer solutions to a^2 + b^2 + c^2 + 2*d^2 = n.

Original entry on oeis.org

1, 6, 14, 20, 30, 40, 36, 48, 62, 42, 72, 100, 68, 120, 112, 48, 126, 108, 98, 180, 136, 160, 180, 144, 132, 126, 216, 200, 240, 280, 112, 192, 254, 120, 252, 320, 210, 360, 324, 144, 264, 252, 288, 420, 340, 280, 336, 288, 260, 342, 294, 360, 408, 520, 360, 240, 496

Offset: 0

N. J. A. Sloane, Feb 15 2014

Seiichi Manyama, Table of n, a(n) for n = 0..10000
I. J. Zucker, Exact Evaluation of Some New Lattice Sums, Symmetry, 2017, 9(12), 314.

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.

Maple
```
See A236924.
```

G.f.: theta_3(q)^3*theta_3(q^2), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

G.f.: 1 + 8*Sum{n >= 1} n*(q^n - q^(3*n) - q^(5*n) + q^(7*n))/(1 - q^(8*n)) - 2*Sum_{n >= 0} (-1)^((n^2+n)/2)*(2*n+1)q^(2*n+1)/(1 - q^(2*n+1)). See Zucker p. 5. Cf. A117000. - Peter Bala, Feb 25 2021

A236926 Number of integer solutions to a^2 + b^2 + c^2 + 3*d^2 = n.

Original entry on oeis.org

1, 6, 12, 10, 18, 48, 40, 12, 60, 78, 24, 48, 70, 84, 120, 32, 66, 192, 84, 36, 144, 180, 120, 96, 136, 126, 168, 82, 84, 336, 200, 60, 252, 288, 96, 96, 234, 228, 360, 140, 120, 480, 144, 84, 336, 336, 240, 192, 310, 258, 252, 128, 252, 624, 400, 96, 408, 540, 168

Offset: 0

N. J. A. Sloane, Feb 15 2014

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
J. Liouville, Sur La Forme x^2+y^2+z^2+3t^2, Journal de Mathématiques Pures et appliquées (2) 8, 193-204 (1863).

Cf. 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.

Maple
```
See A236924.
```

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

A236930 Number of integer solutions to a^2 + b^2 + c^2 + 5*d^2 = n.

Original entry on oeis.org

1, 6, 12, 8, 6, 26, 36, 24, 28, 42, 72, 72, 8, 48, 108, 48, 54, 64, 84, 120, 26, 72, 144, 88, 84, 126, 216, 80, 24, 180, 156, 192, 92, 96, 288, 144, 42, 144, 240, 144, 168, 252, 144, 168, 72, 182, 396, 184, 72, 258, 372, 192, 48, 208, 360, 312, 252, 160, 360, 360, 48

Offset: 0

N. J. A. Sloane, Feb 16 2014

nonn

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

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.

Maple
```
See A236924.
```

Join[{1},Table[Length[Reduce[a^2+b^2+c^2+5d^2==n,{a,b,c,d},Integers]],{n,60}]] (* Harvey P. Dale, Jul 02 2017 *)

G.f.: theta_3(q)^3*theta_3(q^5), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

A236931 Number of integer solutions to a^2 + b^2 + c^2 + 6*d^2 = n.

Original entry on oeis.org

1, 6, 12, 8, 6, 24, 26, 12, 36, 46, 36, 72, 56, 24, 72, 60, 54, 96, 52, 72, 120, 48, 36, 96, 98, 90, 144, 144, 60, 120, 144, 60, 180, 172, 96, 240, 142, 72, 192, 96, 132, 288, 180, 120, 216, 168, 144, 192, 152, 162, 276, 256, 120, 216, 210, 120, 264, 256, 180, 360, 300

Offset: 0

N. J. A. Sloane, Feb 16 2014

nonn

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

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.

Maple
```
See A236924.
```

G.f.: theta_3(q)^3*theta_3(q^6), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

A236932 Number of integer solutions to a^2 + b^2 + c^2 + 7*d^2 = n.

Original entry on oeis.org

1, 6, 12, 8, 6, 24, 24, 2, 24, 54, 40, 36, 56, 72, 48, 24, 66, 96, 84, 40, 72, 144, 24, 12, 120, 102, 120, 80, 98, 132, 72, 64, 84, 240, 160, 48, 198, 180, 120, 72, 136, 240, 240, 84, 84, 312, 120, 96, 248, 246, 180, 96, 216, 228, 240, 80, 156, 360, 216, 120, 168

Offset: 0

N. J. A. Sloane, Feb 16 2014

nonn

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

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.

Maple
```
See A236924.
```

G.f.: theta_3(q)^3*theta_3(q^7), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

A236933 Number of integer solutions to a^2 + b^2 + c^2 + 12*d^2 = n.

Original entry on oeis.org

1, 6, 12, 8, 6, 24, 24, 0, 12, 30, 24, 24, 10, 36, 72, 16, 18, 96, 84, 24, 48, 108, 72, 48, 40, 78, 168, 32, 12, 168, 120, 48, 60, 144, 96, 48, 78, 84, 216, 64, 24, 240, 144, 24, 48, 168, 144, 96, 70, 114, 252, 64, 84, 312, 240, 48, 120, 252, 168, 120, 32

Offset: 0

N. J. A. Sloane, Feb 16 2014

nonn

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

Different from A005875.

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.

Maple
```
See A236924.
```

G.f.: theta_3(q)^3*theta_3(q^12), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 03 2018

A236927 Number of integer solutions to a^2 + b^2 + c^2 + 8*d^2 = n.

Original entry on oeis.org

1, 6, 12, 8, 6, 24, 24, 0, 14, 42, 48, 40, 20, 72, 96, 0, 30, 108, 84, 72, 40, 96, 120, 0, 36, 126, 144, 80, 48, 168, 96, 0, 62, 120, 216, 128, 42, 216, 216, 0, 72, 252, 192, 168, 100, 168, 288, 0, 68, 342, 252, 144, 120, 312, 240, 0, 112, 216, 336, 232, 48, 360, 384

Offset: 0

N. J. A. Sloane, Feb 15 2014

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Liouville, J., Sur la forme X^2 + Y^2 + Z^2 + 8T^2. Journal de mathématiques pures et appliquées 2e série, tome 6 (1861), p. 324-328.

Cf. 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.

Maple
```
See A236924.
```

G.f.: theta_3(q)^3*theta_3(q^8), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

A320149 Number of integer solutions to a^2 + 2b^2 + 2c^2 + 2*d^2 = n.

Original entry on oeis.org

1, 2, 6, 12, 14, 24, 20, 16, 30, 14, 40, 60, 36, 72, 48, 16, 62, 36, 42, 108, 72, 96, 100, 48, 68, 42, 120, 120, 112, 168, 48, 64, 126, 40, 108, 192, 98, 216, 180, 48, 136, 84, 160, 252, 180, 168, 144, 96, 132, 114, 126, 216, 216, 312, 200, 80, 240, 72, 280, 348, 112

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. A320150, A236924, A320151, A320152, A320153.

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

A320150 Number of integer solutions to a^2 + 2b^2 + 2c^2 + 3*d^2 = n.

Original entry on oeis.org

1, 2, 4, 10, 10, 16, 24, 12, 28, 26, 8, 48, 30, 28, 72, 32, 34, 64, 28, 36, 80, 60, 72, 96, 72, 42, 56, 82, 36, 112, 120, 60, 124, 96, 32, 96, 130, 76, 216, 140, 56, 160, 48, 84, 144, 112, 144, 192, 150, 86, 84, 128, 140, 208, 240, 96, 216, 180, 56, 240, 96, 124, 360

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. A320149, A236924, A320151, A320152, A320153.

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

cp's OEIS Frontend

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

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A236928 Number of integer solutions to a^2 + b^2 + c^2 + 2*d^2 = n.

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A236926 Number of integer solutions to a^2 + b^2 + c^2 + 3*d^2 = n.

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A236930 Number of integer solutions to a^2 + b^2 + c^2 + 5*d^2 = n.

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A236931 Number of integer solutions to a^2 + b^2 + c^2 + 6*d^2 = n.

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A236932 Number of integer solutions to a^2 + b^2 + c^2 + 7*d^2 = n.

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Maple

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A236933 Number of integer solutions to a^2 + b^2 + c^2 + 12*d^2 = n.

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Maple

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A236927 Number of integer solutions to a^2 + b^2 + c^2 + 8*d^2 = n.

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Maple

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A320149 Number of integer solutions to a^2 + 2*b^2 + 2*c^2 + 2*d^2 = n.

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A320149 Number of integer solutions to a^2 + 2b^2 + 2c^2 + 2*d^2 = n.

A320150 Number of integer solutions to a^2 + 2b^2 + 2c^2 + 3*d^2 = n.