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

%I A242900 #17 Feb 11 2015 04:10:10
%S A242900 3,10,12,38,56,79,152,251,284,594,920,1108,2136,3402,4407,8350,12863,
%T A242900 17328,33218,52527,70074,133247,214551,294299,547360,883572,1234509,
%U A242900 2284840,3667144,5219161,9551081,15386201,22079741,40061664,64666975,93985744,168363731
%N A242900 Number of compositions of n into exactly two different parts with distinct multiplicities.
%H A242900 Alois P. Heinz, <a href="/A242900/b242900.txt">Table of n, a(n) for n = 4..1000</a>
%F A242900 a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n+1). - _Vaclav Kotesovec_, Aug 21 2014
%e A242900 a(4) = 3: [2,1,1], [1,2,1], [1,1,2].
%e A242900 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].
%p A242900 with(numtheory):
%p A242900 a:= n-> add(add(add(`if`(d<p and (n-p*m)/d<>m, binomial((n-p*m)
%p A242900         /d+m, m), 0), d=divisors(n-p*m)), m=1..n/p), p=2..n-1):
%p A242900 seq(a(n), n=4..60);
%t A242900 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_ *)
%Y A242900 Column k=2 of A242896.
%Y A242900 Cf. A182473 (the same for partitions), A131661 (multiplicities may be equal).
%K A242900 nonn
%O A242900 4,1
%A A242900 _Alois P. Heinz_, May 25 2014