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A348529
Number of compositions (ordered partitions) of n into two or more triangular numbers.
2
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
0,5
FORMULA
a(n) = A023361(n) - A010054(n). - Alois P. Heinz, Oct 21 2021
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Oct 21 2021
STATUS
approved