A348526 -id:A348526 - cp's OEIS frontend

A348528 Number of partitions of n into two or more distinct triangular numbers.

0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 2, 1, 1, 2, 0, 2, 3, 1, 1, 2, 2, 1, 4, 3, 0, 3, 3, 1, 4, 3, 3, 3, 2, 3, 3, 2, 3, 6, 4, 2, 5, 4, 2, 6, 5, 3, 6, 6, 3, 5, 5, 5, 6, 5, 4, 7, 7, 5, 8, 6, 5, 9, 7, 4, 9, 9, 6, 10, 9, 3, 9, 10, 8, 11, 11, 9, 10, 10, 9, 10, 10, 9

Offset: 0

Ilya Gutkovskiy, Oct 21 2021

nonn

Cf. A000217, A010054, A024940, A033985, A348526, A348529.

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

Original entry on oeis.org

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

Ilya Gutkovskiy, Oct 21 2021

nonn

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

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

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 *)

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

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A348528 Number of partitions of n into two or more distinct triangular numbers.

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

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