A025443 Number of partitions of n into 4 distinct nonzero squares.

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

Offset

0,79

Links

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Index entries for sequences related to sums of squares

Formula

a(n) = [x^n y^4] Product_{k>=1} (1 + y*x^(k^2)). - Ilya Gutkovskiy, Apr 22 2019

Maple

b:= proc(n, i, t) option remember; `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))))
    end:
a:= n-> b(n, isqrt(n), 4):
seq(a(n), n=0..150);  # 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, Sqrt[n] // Floor, 4]; Table[a[n], {n, 0, 150}] (* Jean-François Alcover, Feb 29 2016, after Alois P. Heinz*)
dnzs[n_]:=Length[Select[IntegerPartitions[n, {4}], Length[Union[#]]==4&&AllTrue[ Sqrt[ #], IntegerQ] && FreeQ[#, 0]&]]; Array[dnzs, 110, 0] (* Harvey P. Dale, Jun 09 2024 *)

Crossrefs

Cf. A025428 (not necessarily distinct), A025376-A025394 (subsequences), A025417 (greedy inverse).

Column k=4 of A341040.

Sequence in context: A364207 A307791 A307766 * A120080 A227570 A352269

Adjacent sequences: A025440 A025441 A025442 * A025444 A025445 A025446

Keyword

nonn,look

Author

David W. Wilson

Status

approved