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A384671 Expansion of (1-x^2) / (1-2*x-5*x^2+2*x^3).

%I A384671 #9 Aug 31 2025 08:56:40
%S A384671 1,2,8,24,84,272,916,3024,10084,33456,111284,369680,1228868,4083568,
%T A384671 13572116,45104336,149902116,498181680,1655665268,5502434704,
%U A384671 18286832388,60774507760,201978308052,671255490128,2230853504996,7414027844528,24639812233780
%N A384671 Expansion of (1-x^2) / (1-2*x-5*x^2+2*x^3).
%C A384671 Number of walks of length n starting at vertex 0 in the following graph:
%C A384671     1---2
%C A384671    /|\ /|
%C A384671   0 | X |
%C A384671    \|/ \|
%C A384671     4---3.
%H A384671 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384671 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,5,-2).
%F A384671 a(n) = A384673(n)-A384673(n-1). - _R. J. Mathar_, Jun 07 2025
%e A384671 a(2)=8 because we have the walks 0-1-0, 0-1-2, 0-1-3, 0-1-4, 0-4-0, 0-4-1, 0-4-2, 0-4-3.
%p A384671 a:= n-> (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|1>, <0|1|1|0|1>, <1|1|1|1|0>>^n. <<1,1,1,1,1>>)[1,1]:
%p A384671 seq(a(n), n=0..32);
%t A384671 CoefficientList[Series[(1-x^2) / (1-2*x-5*x^2+2*x^3), {x, 0, 32}], x]
%Y A384671 Cf. A384672 (vertices 1 and 4), A384673 (vertices 2 and 3), A384646 (missing edge {2,4}).
%K A384671 nonn,easy,walk,changed
%O A384671 0,2
%A A384671 _Sean A. Irvine_, Jun 05 2025