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A384678 Expansion of (1+x) / (1-2*x-4*x^2+2*x^3).

%I A384678 #9 Jul 07 2025 18:40:50
%S A384678 1,3,10,30,94,288,892,2748,8488,26184,80824,249408,769744,2375472,
%T A384678 7331104,22624608,69822688,215481600,665004736,2052290496,6333636736,
%U A384678 19546425984,60322817920,186164066304,574526552320,1773063734016,5471905544704,16887012920832
%N A384678 Expansion of (1+x) / (1-2*x-4*x^2+2*x^3).
%C A384678 Number of walks of length n starting at vertex 0 in the following graph:
%C A384678      2
%C A384678     /|\
%C A384678  0-1-+-3
%C A384678     \|/
%C A384678      4.
%H A384678 Harvey P. Dale, <a href="/A384678/b384678.txt">Table of n, a(n) for n = 0..1000</a>
%H A384678 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384678 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,4,-2).
%e A384678 a(2)=10 because we have the walks 2-1-0, 2-1-2, 2-1-3, 2-1-4, 2-3-1, 2-3-2, 2-3-4, 2-4-1, 2-4-2, 2-4-3.
%p A384678 a:= n-> (<<0|1|0|0|0>, <1|0|1|1|1>, <0|1|0|1|1>, <0|1|1|0|1>, <0|1|1|1|0>>^n. <<1,1,1,1,1>>)[1,1]:
%p A384678 seq(a(n), n=0..32);
%t A384678 CoefficientList[Series[(1+x) / (1-2*x-4*x^2+2*x^3), {x, 0, 32}], x]
%t A384678 LinearRecurrence[{2,4,-2},{1,3,10},30] (* _Harvey P. Dale_, Jul 07 2025 *)
%Y A384678 Cf. A384677 (vertices 0 and 1), A000244 (missing edge {0,1}), A384633 (missing edge {2,4}), A384640 (missing edge {1,3}).
%K A384678 nonn,easy,walk
%O A384678 0,2
%A A384678 _Sean A. Irvine_, Jun 05 2025