A331843 Number of compositions (ordered partitions) of n into distinct triangular numbers.

1, 1, 0, 1, 2, 0, 1, 2, 0, 2, 7, 2, 0, 2, 6, 1, 4, 6, 2, 12, 24, 3, 8, 0, 8, 32, 6, 2, 13, 26, 6, 34, 36, 0, 32, 150, 3, 20, 50, 14, 54, 126, 32, 32, 12, 55, 160, 78, 122, 44, 174, 4, 72, 294, 36, 201, 896, 128, 62, 180, 176, 164, 198, 852, 110, 320, 159, 212, 414

Offset: 0

Author

Ilya Gutkovskiy, Jan 29 2020

Keywords

• nonn

Examples

			a(10) = 7 because we have [10], [6, 3, 1], [6, 1, 3], [3, 6, 1], [3, 1, 6], [1, 6, 3] and [1, 3, 6].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20000
• Index entries for sequences related to compositions

Crossrefs

Cf. A000217, A023361, A024940, A032020, A032021, A032022, A218396, A219107, A331844, A331845, A331846, A331847.

Programs

Maple

h:= proc(n) option remember; `if`(n<1, 0,
      `if`(issqr(8*n+1), 1+h(n-1), h(n-1)))
    end:
b:= proc(n, i, p) option remember; (t->
      `if`(t*(i+2)/3n, 0, b(n-t, i-1, p+1)))))((i*(i+1)/2))
    end:
a:= n-> b(n, h(n), 0):
seq(a(n), n=0..73);  # Alois P. Heinz, Jan 31 2020

Mathematica

h[n_] := h[n] = If[n<1, 0, If[IntegerQ @ Sqrt[8n+1], 1 + h[n-1], h[n-1]]];
b[n_, i_, p_] := b[n, i, p] = Function[t, If[t (i + 2)/3 < n, 0, If[n == 0, p!, b[n, i-1, p] + If[t>n, 0, b[n - t, i - 1, p + 1]]]]][(i(i + 1)/2)];
a[n_] := b[n, h[n], 0];
a /@ Range[0, 73] (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)