A105039 - OEIS

%I #24 Dec 07 2021 02:27:04

%S 1,1,3,3,8,11,20,34,59,96,167,282,475,800,1352,2275,3828,6426,10785,

%T 18085,30297,50698,84770,141623,236425,394381,657380,1094975,1822628,

%U 3031843,5040129,8373594,13903588,23072567,38267330,63435438,105103059

%N Number of compositions of n with unique smallest part.

%H Vincenzo Librandi, <a href="/A105039/b105039.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: Sum(k*x^(2*k-1)/((1-x^k)*(1-x)^(k-1)), k=1..infinity).

%F Also (1-x)^2*Sum(x^k/(1-x-x^(k+1))^2, k=1..infinity). - _Vladeta Jovovic_, Apr 05 2005

%F a(n) = 1 + sum(k=2..[(n+3)/2], k * sum(s=1..[(n-1)/k], binomial(n-k*s-1, k-2) ) ). - _Max Alekseyev_, Apr 15 2005

%F a(n) ~ (2*sqrt(5)-4)/10 * n * ((1+sqrt(5))/2)^n. - _Vaclav Kotesovec_, May 02 2014

%F Equivalently, a(n) ~ n * phi^(n-3) / 5, where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 07 2021

%e a(5) = 8 because we have 5, 14, 41, 23, 32, 122, 212 and 221.

%p G:= sum(k*x^(2*k-1)/((1-x^k)*(1-x)^(k-1)), k=1..70): Gser:=series(G,x=0,44): seq(coeff(Gser,x^n),n=1..41); # _Emeric Deutsch_, Apr 13 2005

%t nn=37;Drop[CoefficientList[Series[Sum[x^j/(1-x^(j+1)/(1-x))^2,{j,1,nn}],{x,0,nn}],x],1] (* _Geoffrey Critzer_, Mar 31 2014 *)

%o (PARI) a(n)=1+sum(k=2,(n+3)\2,k*sum(s=1,(n-1)\k,binomial(n-k*s-1,k-2))) (Alekseyev)

%Y Cf. A079501, A097979.

%Y Column k=1 of A238342.

%K easy,nonn

%O 1,3

%A _Vladeta Jovovic_, Apr 03 2005

%E More terms from _Emeric Deutsch_ and _Max Alekseyev_, Apr 13 2005