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A384712 Expansion of (1+2*x) / (1-2*x-6*x^2).

%I A384712 #7 Jun 08 2025 05:00:08
%S A384712 1,4,14,52,188,688,2504,9136,33296,121408,442592,1613632,5882816,
%T A384712 21447424,78191744,285068032,1039286528,3788981248,13813681664,
%U A384712 50361250816,183604591616,669376688128,2440380925952,8897021980672,32436329517056,118254790918144
%N A384712 Expansion of (1+2*x) / (1-2*x-6*x^2).
%C A384712 Number of walks of length n in the graph K_{1,1,1,2} starting at vertex 1 when the edges are given by {{0,1}, {0,3}, {0,4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}}.
%C A384712 Also, by symmetry, the number of walks of length n starting at vertex 3 (or 4) in the same graph.
%H A384712 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%F A384712 a(n) = A133592(n+1)/2.
%e A384712 a(2)=14 because we have the walks 1-0-1, 1-0-3, 1-0-4, 1-2-1, 1-2-3, 1-2-4, 1-3-0, 1-3-1, 1-3-2, 1-3-4, 1-4-0, 1-4-1, 1-4-2, 1-4-3.
%p A384712 a:= n-> (<<0|1|0|1|1>, <1|0|1|1|1>, <0|1|0|1|1>, <1|1|1|0|1>, <1|1|1|1|0>>^n. <<1, 1, 1, 1, 1>>)[2, 1]:
%p A384712 seq(a(n), n=0..32);
%t A384712 CoefficientList[Series[(1+2*x) / (1-2*x-6*x^2), {x, 0, 32}], x]
%Y A384712 Cf. A133592, A384711 (vertices 0, 2).
%K A384712 nonn,easy,walk
%O A384712 0,2
%A A384712 _Sean A. Irvine_, Jun 07 2025