A123999 Number of ordered ways of writing n as a sum of 4 squares of nonnegative numbers less than 4.

1, 4, 6, 4, 5, 12, 12, 4, 6, 16, 18, 12, 8, 16, 24, 12, 1, 12, 18, 12, 6, 4, 12, 12, 0, 0, 6, 4, 4, 0, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Jonathan Vos Post, Oct 31 2006

Keywords

• easy
• nonn

Comments

Through n = 15, a(n) = number of ordered ways to write n as the sum of 4 squares. For n > 15, we must exclude sums which include 4^2, 5^2, 6^2 and the like. The values of n such that a(n) = 0 are 16, 24, 25, 29, 30, 32, 33, 34, 35 and all n > 36. Without the restriction on the size of squares, all natural numbers can be written as the sum of 4 squares, as Lagrange proved in 1750. This sequence is to 4 as A123337 Number of ordered ways to write n as the sum of 5 squares less than 5, is to 5.

Examples

			a(0) = 1 because of the unique sum 0 = 0^2 + 0^2 + 0^2 + 0^2.
a(1) = 4 because of the 4 permutations 1 = 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(4) = 5 because of 4 = 1^2 + 1^2 + 1^2 + 1^2 plus the 4 permutations of 4 = 0^2 + 0^2 + 0^2 + 2^2.
a(16) = 1 because 16 = 2^2 + 2^2 + 2^2 + 2^2.
		

Crossrefs

Cf. A000118, A014110, A123337.

Programs

Mathematica

a[n_] := Total[ Length /@ Permutations /@ IntegerPartitions[n, {4}, Range[0, 3]^2]]; a /@ Range[0, 72] (* Giovanni Resta, Jun 13 2016 *)

Formula

a(n) = Card{(a,b,c,d) such that 0<=a,b,c,d<4 and a^2 + b^2 + c^2 + d^2 = n}.

Extensions

Corrected typo in third example Dave Zobel (dzobel(AT)alumni.caltech.edu), Mar 07 2009

a(16) and related example corrected by Giovanni Resta, Jun 13 2016