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

Author

Alois P. Heinz, May 25 2014

Keywords

• nonn

Examples

			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].
		

Links

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

Crossrefs

Column k=2 of A242896.

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

Programs

Maple

with(numtheory):
a:= n-> add(add(add(`if`(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
Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

Formula

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