This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A105039 #32 May 13 2025 18:29:32
%S A105039 1,1,3,3,8,11,20,34,59,96,167,282,475,800,1352,2275,3828,6426,10785,
%T A105039 18085,30297,50698,84770,141623,236425,394381,657380,1094975,1822628,
%U A105039 3031843,5040129,8373594,13903588,23072567,38267330,63435438,105103059,174054820,288105394
%N A105039 Number of compositions of n with unique smallest part.
%H A105039 Alois P. Heinz, <a href="/A105039/b105039.txt">Table of n, a(n) for n = 1..2000</a> (first 1000 terms from Vincenzo Librandi)
%F A105039 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.
%F A105039 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 A105039 a(n) ~ (2*sqrt(5)-4)/10 * n * ((1+sqrt(5))/2)^n. - _Vaclav Kotesovec_, May 02 2014
%F A105039 Equivalently, a(n) ~ n * phi^(n-3) / 5, where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 07 2021
%e A105039 a(5) = 8 because we have 5, 14, 41, 23, 32, 122, 212 and 221.
%p A105039 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 A105039 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 A105039 (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 A105039 Cf. A079501, A097979.
%Y A105039 Column k=1 of A238342.
%K A105039 easy,nonn
%O A105039 1,3
%A A105039 _Vladeta Jovovic_, Apr 03 2005
%E A105039 More terms from _Emeric Deutsch_ and _Max Alekseyev_, Apr 13 2005