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

0,2

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.

Links

Table of n, a(n) for n=0..72.

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}.

Example

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.

Mathematica

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

Crossrefs

Cf. A000118, A014110, A123337.

Sequence in context: A132024 A092039 A243371 * A014110 A266491 A323016

Adjacent sequences: A123996 A123997 A123998 * A124000 A124001 A124002

Keyword

easy,nonn

Author

Jonathan Vos Post, Oct 31 2006

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

Status

approved