A348529 Number of compositions (ordered partitions) of n into two or more triangular numbers.

0, 0, 1, 1, 3, 4, 6, 11, 16, 25, 39, 61, 94, 147, 227, 350, 546, 846, 1309, 2030, 3147, 4875, 7558, 11715, 18154, 28136, 43609, 67586, 104747, 162346, 251610, 389958, 604381, 936699, 1451743, 2249991, 3487152, 5404570, 8376292, 12982016, 20120202, 31183350, 48329596, 74903735

Offset: 0

Author

Ilya Gutkovskiy, Oct 21 2021

Keywords

• nonn

Crossrefs

Cf. A000217, A010054, A023361, A347805, A348526, A348528.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(
     `if`(issqr(8*j+1), b(n-j), 0), j=1..n))
    end:
a:= n-> b(n)-`if`(issqr(8*n+1), 1, 0):
seq(a(n), n=0..43);  # Alois P. Heinz, Oct 21 2021

Mathematica

b[n_] := b[n] = If[n == 0, 1, Sum[
     If[IntegerQ@ Sqrt[8*j + 1], b[n - j], 0], {j, 1, n}]];
a[n_] := b[n] - If[IntegerQ@ Sqrt[8*n + 1], 1, 0];
Table[a[n], {n, 0, 43}] (* Jean-François Alcover, Mar 01 2022, after Alois P. Heinz *)

Formula

a(n) = A023361(n) - A010054(n). - Alois P. Heinz, Oct 21 2021