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

0,5

Links

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

Index entries for sequences related to compositions

Example

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].

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)/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))
    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 *)

Crossrefs

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

Sequence in context: A309953 A281666 A049803 * A330635 A322378 A053121

Adjacent sequences: A331840 A331841 A331842 * A331844 A331845 A331846

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 29 2020

Status

approved