cp's OEIS Frontend

A384633 Expansion of (1+x-2*x^2-2*x^3) / (1-6*x^2-4*x^3+2*x^4).

%I A384633 #8 Jul 08 2025 11:48:04
%S A384633 1,1,4,8,26,62,180,460,1276,3356,9136,24320,65688,175752,473136,
%T A384633 1268624,3410448,9152784,24590912,66021248,177335712,476185568,
%U A384633 1278917440,3434413760,9223575488,24769781184,66521273088,178644161536,479759612288,1288410499200
%N A384633 Expansion of (1+x-2*x^2-2*x^3) / (1-6*x^2-4*x^3+2*x^4).
%C A384633 Number of walks of length n starting at vertex 0 in the following graph:
%C A384633      2
%C A384633     / \
%C A384633  0-1---3
%C A384633     \ /
%C A384633      4.
%C A384633 Also, for n>=1, the number of walks of length n-1 starting at vertex 1 in the same graph.
%H A384633 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384633 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,4,-2).
%e A384633 a(3)=8 because we have the walks 0-1-0-1, 0-1-2-1, 0-1-2-3, 0-1-3-1, 0-1-3-2, 0-1-3-4, 0-1-4-1, 0-1-4-3.
%p A384633 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>>)[1,1]:
%p A384633 seq(a(n), n=0..32);
%t A384633 CoefficientList[Series[(1+x-2*x^2-2*x^3) / (1-6*x^2-4*x^3+2*x^4), {x, 0, 32}], x]
%Y A384633 Cf. A384634 (vertices 2, 4), A384635 (vertex 3), A005824 (missing edge {1,3}), A105476 (missing edge {1,2}).
%K A384633 nonn,easy,walk
%O A384633 0,3
%A A384633 _Sean A. Irvine_, Jun 05 2025