A175958 Number of partitions of n^2 into 4 distinct nonzero squares.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 5, 0, 4, 4, 5, 0, 10, 4, 7, 0, 11, 7, 17, 1, 13, 17, 15, 0, 29, 13, 27, 4, 23, 17, 41, 0, 29, 35, 32, 4, 66, 24, 38, 0, 47, 35, 73, 7, 50, 56, 73, 1, 91, 42, 63, 17, 68, 49, 125, 0, 103, 93, 83, 13, 133, 86, 93, 4

Offset: 0

Author

R. J. Mathar, Oct 30 2010

Keywords

• nonn
• look

Examples

			a(9)=1 refers to the partition 9^2 = 2^2+4^2+5^2+6^2. a(11)=1 refers to 11^2 = 1^2+2^2+4^2+10^2. a(13)=2 refers to 13^2 = 1^2+2^2+8^2+10^2 = 2^2+4^2+7^2+10^2.
		

Links

• Alois P. Heinz and Donovan Johnson, Table of n, a(n) for n = 0..1000 (terms up to a(300) from Alois P. Heinz)

Crossrefs

Cf. A004195, A004441, A004433, A181524.

Programs

Maple

A025443 := proc(n) local res,a,b,c,d ; res := 0 ; for a from 1 do if 4*a^2 > n then break; fi; for b from a+1 do if a^2+3*b^2 > n then break; fi; for c from b+1 do if a^2+b^2+2*c^2 > n then break; fi; for d from c+1 do if a^2+b^2+c^2+d^2 > n then break; elif a^2+b^2+c^2+d^2 = n then res := res+1 ; fi ; end do; end do; end do: end do: res ; end proc:
A := proc(n) A025443(n^2) ; end proc: seq(A(n),n=0..60) ;
# second Maple program:
b:= proc(n,i,t) option remember; `if`(n=0, `if`(t=0,1,0),
      `if`(t*i^2n, 0, b(n-i^2,i-1,t-1))))
    end:
a:= n-> b(n^2, n, 4):
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 07 2013

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[t*i^2 < n, 0, If[i == 1, 0, b[n, i-1, t]] + If[i^2 > n, 0, b[n-i^2, i-1, t-1]]]]; a[n_] := b[n^2, n, 4]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 24 2015, after Alois P. Heinz *)

Formula

a(n) = A025443(n^2).

Extensions

More terms from Alois P. Heinz, Feb 07 2013