A025442 Number of partitions of n into 3 distinct nonzero squares.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 1, 0, 1, 1, 0, 2, 0, 0, 2, 2, 1, 0, 1, 2, 0, 0, 0, 2, 0, 0, 3, 0, 0, 1, 2, 1, 1

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A024803, A025339, A001974, A004432.

Column k=3 of A341040.

Programs

Maple

b:= proc(n,i,t) option remember; `if`(n=0, `if`(t=0,1,0),
      `if`(i<1 or t<1, 0, `if`(i=1, 0, b(n,i-1,t))+
      `if`(i^2>n, 0, b(n-i^2,i-1,t-1))))
    end:
a:= n-> b(n, isqrt(n), 3):
seq(a(n), n=0..120);  # Alois P. Heinz, Feb 07 2013

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t==0, 1, 0], If[i<1 || t<1, 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, Sqrt[n] // Floor, 3]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Oct 10 2015, after Alois P. Heinz *)

PARI

A025442(n)={sum(x=1,sqrtint(n\3),sum(y=x+1,sqrtint((n-1-x^2)\2),issquare(n-x^2-y^2)))} \\ - M. F. Hasler, Feb 03 2013

Formula

a(n)>0 <=> n is in A004432. - M. F. Hasler, Feb 03 2013

a(n) = [x^n y^3] Product_{k>=1} (1 + y*x^(k^2)). - Ilya Gutkovskiy, Apr 22 2019