A357354 Number of partitions of n into distinct positive squares such that the number of parts is a square.

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

Offset: 0

Author

Ilya Gutkovskiy, Sep 25 2022

Keywords

• nonn
• look
• changed

Examples

			a(30) = 1 because we have [16,9,4,1].
a(78) = 3: [36,25,16,1], [49,16,9,4], [64,9,4,1].
		

Links

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

Crossrefs

Cf. A000290, A033461, A045450, A089333, A357352.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0,
     `if`(issqr(t), 1, 0), `if`(n>i*(i+1)*(2*i+1)/6, 0,
     `if`(i^2>n, 0, b(n-i^2, i-1, t+1))+b(n, i-1, t)))
    end:
a:= n-> b(n, isqrt(n), 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 25 2022

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0,
   If[IntegerQ @ Sqrt[t], 1, 0], If[n > i*(i+1)*(2*i+1)/6, 0,
   If[i^2 > n, 0, b[n-i^2, i-1, t+1]] + b[n, i-1, t]]];
a[n_] := b[n, Floor @ Sqrt[n], 0];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 24 2025, after Alois P. Heinz *)