A331884 Number of compositions (ordered partitions) of n^2 into distinct squares.

1, 1, 1, 1, 1, 3, 1, 7, 1, 31, 123, 151, 121, 897, 7351, 5415, 14881, 48705, 150583, 468973, 1013163, 1432471, 1730023, 50432107, 14925241, 125269841, 74592537, 241763479, 213156871, 895153173, 7716880623, 2681163865, 3190865761, 22501985413, 116279718801

Offset

0,6

Links

Alois P. Heinz, Table of n, a(n) for n = 0..300

Index entries for sequences related to compositions

Index entries for sequences related to sums of squares

Formula

a(n) = A331844(A000290(n)).

Example

a(5) = 3 because we have [25], [16, 9] and [9, 16].

Maple

b:= proc(n, i, p) option remember;
      `if`(i*(i+1)*(2*i+1)/6<n, 0, `if`(n=0, p!,
      `if`(i^2>n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n^2, n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 30 2020

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)(2i+1)/6 < n, 0, If[n == 0, p!, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]];
a[n_] := b[n^2, n, 0];
a /@ Range[0, 35] (* Jean-François Alcover, Nov 08 2020, after Alois P. Heinz *)

Crossrefs

Cf. A000290, A006456, A030273, A032020, A037444, A105152, A224366, A232173, A280129, A298640, A331844.

Sequence in context: A352402 A278954 A350802 * A146430 A146281 A146265

Adjacent sequences: A331881 A331882 A331883 * A331885 A331886 A331887

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 30 2020

Extensions

a(24)-a(34) from Alois P. Heinz, Jan 30 2020

Status

approved