A014110 Number of ordered ways of writing n as a sum of 4 squares of nonnegative integers.

1, 4, 6, 4, 5, 12, 12, 4, 6, 16, 18, 12, 8, 16, 24, 12, 5, 24, 30, 16, 18, 28, 24, 12, 12, 28, 42, 28, 12, 36, 48, 16, 6, 36, 42, 36, 29, 28, 48, 28, 18, 48, 60, 28, 24, 60, 48, 24, 8, 44, 72, 48, 30, 48, 84, 36, 24, 52, 54, 48, 36, 52, 72, 52, 5, 72, 96, 40, 42, 72

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• easy
• nonn

Comments

This counts ordered sums of squares of nonnegative integers, whereas A000118 counts ordered sums of squares of integers of any sign. - R. J. Mathar, May 16 2023

Examples

			From  _R. J. Mathar_, May 16 2023: (Start)
a(1)=4 counts 0^2+0^2+0^2+1^2 = 0^2+0^2+1^2+0^2 = 0^2+1^2+0^2+0^2 = 1^2+0^2+0^2+0^2.
a(2)=6 counts 0^2+0^2+1^2+1^2 = 0^2+1^2+0^2+1^2 = 0^2+1^2+1^2+0^2 = 1^2+0^2+0^2+1^2 = 1^2+0^2+1^2+0^2 = 1^2+1^2+0^2+0^2. (End)
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to sums of squares

Crossrefs

Convolution square of A000925.

Programs

Mathematica

a = Compile[{{n, Integer}}, Block[{c = 0, k, j, i = Floor[Sqrt[n]]}, While[i > -1, j = Floor[Sqrt[n - i^2]]; While[j > -1, k = Floor[Sqrt[n - i^2 - j^2]]; While[k > -1, c += Boole[ Mod[ Sqrt[ n - i^2 - j^2 - k^2], 1] == 0]; k--]; j--]; i--]; c]]; Array[a, 70, 0] - (* _Robert G. Wilson v, Aug 13 2025 *)

Formula

Coefficient of q^n in (1/16)*(1 + theta_3(0, q))^4; or coeff. of q^n in (Sum q^(i^2), i=0..inf)^4.