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A374259 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3), with a(0)=4, a(1)=6, a(2)=20.

%I A374259 #19 Jul 21 2024 12:48:10
%S A374259 4,6,20,48,130,336,884,2310,6052,15840,41474,108576,284260,744198,
%T A374259 1948340,5100816,13354114,34961520,91530452,239629830,627359044,
%U A374259 1642447296,4299982850,11257501248,29472520900,77160061446,202007663444,528862928880,1384581123202,3624880440720,9490060198964
%N A374259 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3), with a(0)=4, a(1)=6, a(2)=20.
%C A374259 a(n) is the number of edge covers of a rocket graph R_{3,n,n}.
%C A374259 A rocket graph R_{3,n,n} is cycle graph C_3 with two paths of n edges, where an end vertex of each path is identified with a distinct vertex in the C_3.
%C A374259 In other words, a rocket graph is a path with vertices -n-1, ..., -1, 0, 1, ..., n+1 with an additional edge (-1,1).
%H A374259 Paolo Xausa, <a href="/A374259/b374259.txt">Table of n, a(n) for n = 0..1000</a>
%H A374259 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A374259 G.f.: (4-2*x)/(1-2*x-2*x^2+x^3).
%F A374259 a(n) = 2*A059929(n+1).
%F A374259 a(n) = Fibonacci(2n+2)+3*Fibonacci(n+1)*Fibonacci(n+1).
%e A374259 For n=1, the R_{3,1,1} rocket graph is as follows and has a(1)=6 edge covers.
%e A374259     *--*
%e A374259    /|
%e A374259   * |
%e A374259    \|
%e A374259     *--*
%t A374259 LinearRecurrence[{2, 2, -1}, {4, 6, 20}, 50] (* _Paolo Xausa_, Jul 20 2024 *)
%Y A374259 Equals twice A059929.
%K A374259 nonn,easy
%O A374259 0,1
%A A374259 _Bridget Rozema_, Jul 01 2024