A105039 Number of compositions of n with unique smallest part.
1, 1, 3, 3, 8, 11, 20, 34, 59, 96, 167, 282, 475, 800, 1352, 2275, 3828, 6426, 10785, 18085, 30297, 50698, 84770, 141623, 236425, 394381, 657380, 1094975, 1822628, 3031843, 5040129, 8373594, 13903588, 23072567, 38267330, 63435438, 105103059, 174054820, 288105394
Offset: 1
Examples
a(5) = 8 because we have 5, 14, 41, 23, 32, 122, 212 and 221.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..2000 (first 1000 terms from Vincenzo Librandi)
Programs
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Maple
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
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Mathematica
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 *) -
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)
Formula
G.f.: Sum_{k>0} k * x^(2*k-1)/((1 - x^k) * (1 - x)^(k-1)) = (1 - x)^2 * Sum_{k>0} x^k/(1 - x - x^(k+1))^2.
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
a(n) ~ (2*sqrt(5)-4)/10 * n * ((1+sqrt(5))/2)^n. - Vaclav Kotesovec, May 02 2014
Equivalently, a(n) ~ n * phi^(n-3) / 5, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021
Extensions
More terms from Emeric Deutsch and Max Alekseyev, Apr 13 2005