A123337 Number of ordered ways to write n as the sum of 5 squares less than 5^2.

1, 5, 10, 10, 10, 21, 30, 20, 15, 35, 50, 40, 30, 45, 70, 60, 30, 55, 100, 80, 56, 90, 110, 80, 60, 85, 120, 100, 60, 90, 130, 80, 35, 90, 120, 80, 65, 85, 90, 60, 35, 60, 90, 50, 30, 61, 60, 20, 10, 50, 40, 30, 25, 20, 30, 0, 10, 20, 20, 10, 0, 20, 0, 0, 5, 5, 10, 0, 5, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Jonathan Vos Post, Oct 11 2006

Keywords

• easy
• fini
• full
• nonn

Comments

Through n = 24, a(n) = number of ordered ways to write n as the sum of 5 squares. For n > 24, we must exclude sums which include 5^2, 6^2 and the like. The values of n such that a(n) = 0 are 55, 60, 62, 63, 67, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79 and all n > 80. 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.

Examples

			a(0) = 1 because the unique such sum is 0 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2.
a(1) = 5 because there are 5 permutations of 1 = 1^2 + 0^2 + 0^2 + 0^2 + 0^2, such as 1 = 0^2 + 1^2 + 0^2 + 0^2 + 0^2.
a(2) = 10 because there are 10 permutations of 2 = 1^2 + 1^2 + 0^2 + 0^2 + 0^2, such as 2 = 1^2 + 0^2 + 1^2 + 0^2 + 0^2.
a(5) = 21 because of the unique sum 5 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 and also 20 permutations of 5 = 2^2 + 1^2 + 0^2 + 0^2 + 0^2.
a(16) = 30 because there are 5 permutations of 16 = 4^2 + 0^2 + 0^2 + 0^2 + 0^2 and 5 permutations of 16 = 0^2 + 2^2 + 2^2 + 2^2 + 2^2 and 20 permutations of 16 = 3^2 + 2^2 + 1^2 + 1^2 + 1^2.
		

Crossrefs

Cf. A000118, A014110.

Programs

Mathematica

a[n_] := Total[ Length /@ Permutations /@ IntegerPartitions[n, {5}, Range[0, 4]^2]]; a /@ Range[0, 80] (* Giovanni Resta, Jun 13 2016 *)

Extensions

23 terms corrected by Giovanni Resta, Jun 13 2016