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A384614 Expansion of (1+x+x^2) / (1-x-3*x^2).

%I A384614 #5 Jun 04 2025 23:20:44
%S A384614 1,2,6,12,30,66,156,354,822,1884,4350,10002,23052,53058,122214,281388,
%T A384614 648030,1492194,3436284,7912866,18221718,41960316,96625470,222506418,
%U A384614 512382828,1179902082,2717050566,6256756812,14407908510,33178178946,76401904476,175936441314
%N A384614 Expansion of (1+x+x^2) / (1-x-3*x^2).
%C A384614 Number of walks of length n starting at vertex 2 in the following graph:
%C A384614   0-1
%C A384614     |\
%C A384614     | 2
%C A384614     |/
%C A384614   4-3.
%H A384614 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%e A384614 a(2)=6 because we have the walks 2-1-0, 2-1-2, 2-1-3, 2-3-1, 2-3-2, 2-3-4.
%p A384614 a:= n-> (<<0|1|0|0|0>, <1|0|1|1|0>, <0|1|0|1|0>, <0|1|1|0|1>, <0|0|0|1|0>>^n. <<1,1,1,1,1>>)[3,1]:
%p A384614 seq(a(n), n=0..32);
%t A384614 CoefficientList[Series[(1+x+x^2) / (1-x-3*x^2), {x, 0, 32}], x]
%Y A384614 Cf. A105476 (vertices 0, 1), A382683 (missing edge {4,3}).
%K A384614 nonn,easy,walk
%O A384614 0,2
%A A384614 _Sean A. Irvine_, Jun 04 2025