cp's OEIS Frontend

A099234 A trisection of 1/(1-x-x^4).

%I A099234 #31 May 20 2024 11:25:00
%S A099234 1,1,4,10,26,69,181,476,1252,3292,8657,22765,59864,157422,413966,
%T A099234 1088589,2862617,7527704,19795288,52054840,136886433,359964521,
%U A099234 946583628,2489191330,6545722210,17213011605,45264335853,119029728628
%N A099234 A trisection of 1/(1-x-x^4).
%C A099234 A row of A099233.
%C A099234 Row sums of number triangle A116089. - _Paul Barry_, Feb 04 2006
%H A099234 Michael De Vlieger, <a href="/A099234/b099234.txt">Table of n, a(n) for n = 0..2382</a>
%H A099234 Hùng Việt Chu, Nurettin Irmak, Steven J. Miller, László Szalay, and Sindy Xin Zhang, <a href="https://arxiv.org/abs/2304.05409">Schreier Multisets and the s-step Fibonacci Sequences</a>, arXiv:2304.05409 [math.CO], 2023. See also <a href="https://math.colgate.edu/~integers/a7Proc23/a7Proc23.pdf">Integers</a> (2024) Vol. 24A, Art. No. A7, p. 3.
%H A099234 Milan Janjić, <a href="https://www.emis.de/journals/JIS/VOL21/Janjic2/janjic103.html">Pascal Matrices and Restricted Words</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
%H A099234 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,3,1).
%F A099234 G.f.: 1/(1-x*(1+x)^3).
%F A099234 a(n) = Sum_{k=0..n} binomial(3*(n-k), k).
%F A099234 a(n) = a(n-1)+3*a(n-2)+3*a(n-3)+a(n-4).
%F A099234 a(n) = A003269(3n).
%F A099234 a(n) = Sum_{k=0..n} C(3*k,n-k) = Sum_{k=0..n} C(n,k)*C(4*k,n)/C(4*k,k). - _Paul Barry_, Feb 04 2006
%F A099234 G.f.: 1/(G(0) - x) where G(k) = 1 - (2*k+3)*x/(2*k+1 - x*(k+2)*(2*k+1)/(x*(k+2) - (k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 23 2012
%t A099234 CoefficientList[Series[1/(1-x (1+x)^3),{x,0,30}],x] (* or *) LinearRecurrence[{1,3,3,1},{1,1,4,10},30] (* _Harvey P. Dale_, Jun 05 2011 *)
%Y A099234 Cf. A003269, A099233, A116089.
%K A099234 easy,nonn
%O A099234 0,3
%A A099234 _Paul Barry_, Oct 08 2004