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A384642 Expansion of (1+x-x^2+x^3) / (1-x-5*x^2+x^3+2*x^4).

%I A384642 #6 Jul 08 2025 11:55:01
%S A384642 1,2,6,16,42,112,294,780,2054,5436,14338,37904,100050,264360,698030,
%T A384642 1843972,4869662,12862772,33971050,89727304,236980458,625920384,
%U A384642 1653153270,4366320124,11532205174,30458811756,80447210962,212476424320,561189257026,1482206544152
%N A384642 Expansion of (1+x-x^2+x^3) / (1-x-5*x^2+x^3+2*x^4).
%C A384642 Number of walks of length n starting at vertex 3 in the following graph:
%C A384642      2
%C A384642     /|\
%C A384642  0-1 | 3
%C A384642     \|/
%C A384642      4.
%H A384642 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384642 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-1,-2).
%e A384642 a(2)=6 because we have the walks 3-2-1, 3-2-3, 3-2-4, 3-4-1, 3-4-2, 3-4-3.
%p A384642 a:= n-> (<<0|1|0|0|0>, <1|0|1|0|1>, <0|1|0|1|1>, <0|0|1|0|1>, <0|1|1|1|0>>^n. <<1,1,1,1,1>>)[4,1]:
%p A384642 seq(a(n), n=0..32);
%t A384642 CoefficientList[Series[(1+x-x^2+x^3) / (1-x-5*x^2+x^3+2*x^4), {x, 0, 32}], x]
%Y A384642 Cf. A384640 (vertices 0, 1), A384641 (vertex 2), A005824 (missing edge {2,4}), A026597 (missing edge {0,1}).
%K A384642 nonn,easy,walk
%O A384642 0,2
%A A384642 _Sean A. Irvine_, Jun 05 2025