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A384635 Expansion of (1+3*x+2*x^2) / (1-6*x^2-4*x^3+2*x^4).

%I A384635 #6 Jul 08 2025 11:50:39
%S A384635 1,3,8,22,58,158,420,1136,3036,8180,21920,58952,158168,425032,1140976,
%T A384635 3064960,8229648,22103600,59355776,159410272,428089760,1149677536,
%U A384635 3087468096,8291603712,22267339200,59800139584,160595513856,431286986880,1158238963072
%N A384635 Expansion of (1+3*x+2*x^2) / (1-6*x^2-4*x^3+2*x^4).
%C A384635 Number of walks of length n starting at vertex 3 in the following graph:
%C A384635      2
%C A384635     / \
%C A384635  0-1---3
%C A384635     \ /
%C A384635      4.
%H A384635 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384635 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,4,-2).
%e A384635 a(2)=8 because we have the walks 3-1-0, 3-1-2, 3-1-3. 3-1-4, 3-2-1, 3-2-3, 3-4-1, 3-4-3.
%p A384635 a:= n-> (<<0|1|0|0|0>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <0|1|0|1|0>>^n. <<1,1,1,1,1>>)[3,1]:
%p A384635 seq(a(n), n=0..32);
%t A384635 CoefficientList[Series[(1+2*x+x^2) / (1-6*x^2-4*x^3+2*x^4), {x, 0, 32}], x]
%Y A384635 Cf. A384633 (vertices 0, 1), A384634 (vertices 2, 4), A005824 (missing edge {1,3}), A105476 (missing edge {1,2}).
%K A384635 nonn,easy,walk
%O A384635 0,2
%A A384635 _Sean A. Irvine_, Jun 05 2025