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

0,2

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

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

Index entries for sequences related to sums of squares

Formula

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

Example

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)

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 *)

Crossrefs

Convolution square of A000925.

Sequence in context: A092039 A243371 A123999 * A266491 A323016 A261637

Adjacent sequences: A014107 A014108 A014109 * A014111 A014112 A014113

Keyword

easy,nonn

Author

Joe Keane (jgk(AT)jgk.org)

Status

approved