A242900 Number of compositions of n into exactly two different parts with distinct multiplicities.

3, 10, 12, 38, 56, 79, 152, 251, 284, 594, 920, 1108, 2136, 3402, 4407, 8350, 12863, 17328, 33218, 52527, 70074, 133247, 214551, 294299, 547360, 883572, 1234509, 2284840, 3667144, 5219161, 9551081, 15386201, 22079741, 40061664, 64666975, 93985744, 168363731

Offset

4,1

Links

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Formula

a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n+1). - Vaclav Kotesovec, Aug 21 2014

Example

a(4) = 3: [2,1,1], [1,2,1], [1,1,2].
a(5) = 10: [2,1,1,1], [1,2,1,1], [1,1,2,1], [1,1,1,2], [2,2,1], [2,1,2], [1,2,2], [3,1,1], [1,3,1], [1,1,3].

Maple

with(numtheory):
a:= n-> add(add(add(`if`(d<p and (n-p*m)/d<>m, binomial((n-p*m)
        /d+m, m), 0), d=divisors(n-p*m)), m=1..n/p), p=2..n-1):
seq(a(n), n=4..60);

Mathematica

div[0] = {}; div[n_] := Divisors[n]; a[n_] := Sum[Sum[Sum[If[d<p && (n-p*m)/d != m, Binomial[(n-p*m)/d+m, m], 0], {d, div[n-p*m]}], {m, 1, n/p}], {p, 2, n-1}]; Table[ a[n], {n, 4, 60}] (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

Crossrefs

Column k=2 of A242896.

Cf. A182473 (the same for partitions), A131661 (multiplicities may be equal).

Sequence in context: A102017 A032916 A044994 * A358266 A299403 A174242

Adjacent sequences: A242897 A242898 A242899 * A242901 A242902 A242903

Keyword

nonn

Author

Alois P. Heinz, May 25 2014

Status

approved