A339431 Number of compositions (ordered partitions) of n into an odd number of distinct squares.

0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, 0, 1, 0, 0, 0, 0, 6, 0, 0, 0, 1, 6, 0, 0, 6, 6, 0, 0, 0, 0, 6, 1, 0, 6, 0, 0, 6, 6, 0, 0, 6, 6, 0, 0, 7, 6, 0, 0, 6, 6, 120, 6, 0, 0, 6, 0, 6, 12, 0, 1, 6, 126, 0, 0, 12, 6, 0, 0, 0, 12, 126, 0, 12, 6, 120, 0, 7

Offset: 0

Author

Ilya Gutkovskiy, Dec 04 2020

Keywords

• nonn
• look

Examples

			a(55) = 120 because we have [25, 16, 9, 4, 1] (120 permutations).
		

Links

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

Crossrefs

Cf. A000290, A331844, A332304, A339367, A339419, A339430.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, irem(p, 2)*p!,
     (s-> `if`(s>n, 0, b(n, i+1, p)+b(n-s, i+1, p+1)))(i^2))
    end:
a:= n-> b(n, 1, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 04 2020

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, Mod[p, 2]*p!, With[{s = i^2}, If[s > n, 0, b[n, i + 1, p] + b[n - s, i + 1, p + 1]]]];
a[n_] := b[n, 1, 0];
a /@ Range[0, 100] (* Jean-François Alcover, Mar 14 2021, after Alois P. Heinz *)