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 A099134 #20 Feb 23 2024 07:29:28
%S A099134 0,1,2,23,84,605,2806,17107,87528,500089,2663210,14828111,80257212,
%T A099134 442248533,2409384094,13221490315,72221278416,395650872817,
%U A099134 2163506035538,11844378654599,64795371984420,354633938406221
%N A099134 Expansion of x/(1-2x-19x^2).
%C A099134 Binomial transform is A099133. Binomial transform of x/(1-20x^2), or (0,1,0,20,0,400,0,8000,....). The inverse binomial transform of k^(n-1)Fib(n) has g.f. x/(1-(k-2)x-(k^2+k-1)x^2).
%C A099134 4*a(n) = (-1)^(n+1)*b(n;4) = 3^n*b(n;4/3), where b(n;d), n=0,1,..., d \in C, denote one of the delta-Fibonacci numbers defined in comments to A014445 (see also Witula-Slota's paper). Our first identity is equivalent to the second formula given below. We note that the sequence (4/3)^n*F(n) is the binomial transform of the sequence 3^(-n)*b(n;4). - Roman Witula, Jul 24 2012
%D A099134 R. Witula, D. Slota, \delta-Fibonacci Numbers, Appl. Anal. Discrete Math., 3 (2009), 310-329.
%H A099134 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,19).
%F A099134 a(n) = 2a(n-1) + 19a(n-2).
%F A099134 a(n) = sum{k=0..n, (-1)^(n-k)binomial(n, k)4^(k-1)*Fib(k)}.
%F A099134 a(n) = sum{k=0..n, binomial(n, 2k+1)20^k}.
%t A099134 Join[{a=0,b=1},Table[c=2*b+19*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 01 2011 *)
%t A099134 CoefficientList[Series[x/(1-2x-19x^2),{x,0,30}],x] (* or *) LinearRecurrence[ {2,19},{0,1},30] (* _Harvey P. Dale_, Dec 25 2019 *)
%Y A099134 Cf. A015447.
%K A099134 easy,nonn
%O A099134 0,3
%A A099134 _Paul Barry_, Sep 29 2004