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A384600 Expansion of (1+x-x^2) / (1-x-4*x^2+2*x^3+2*x^4).

%I A384600 #12 Jun 04 2025 17:46:06
%S A384600 1,2,5,11,25,55,123,271,603,1331,2955,6531,14483,32035,70995,157107,
%T A384600 348051,770419,1706419,3777779,8366515,18523955,41021619,90828851,
%U A384600 201134387,445358643,986195251,2183703347,4835498291,10707203891,23709399859,52499812147
%N A384600 Expansion of (1+x-x^2) / (1-x-4*x^2+2*x^3+2*x^4).
%C A384600 Number of walks of length n on the following graph starting at vertex 3:
%C A384600            3
%C A384600           /|
%C A384600      0-1-2 |
%C A384600           \|
%C A384600            4.
%H A384600 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384600 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-2,-2).
%e A384600 a(2)=5 because we have the walk 3-2-1, 3-2-3, 3-2-4, 3-4-2, 3-4-3.
%p A384600 a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-2|-2|4|1>>^n. <<1,2,5,11>>)[1,1]:
%p A384600 seq(a(n), n=0..31);  # _Alois P. Heinz_, Jun 04 2025
%t A384600 CoefficientList[Series[(1 + x - x^2)/(1 - x - 4*x^2 + 2*x^3 + 2*x^4), {x, 0, 31}], x] (* _Michael De Vlieger_, Jun 04 2025 *)
%Y A384600 Cf. A384598 (vertices 0 and 1), A384599 (vertex 2), A062112 (missing edge {3,4}), A382683 (missing edge {0,1}).
%K A384600 nonn,walk,easy
%O A384600 0,2
%A A384600 _Sean A. Irvine_, Jun 04 2025