A384712 Expansion of (1+2*x) / (1-2*x-6*x^2).

1, 4, 14, 52, 188, 688, 2504, 9136, 33296, 121408, 442592, 1613632, 5882816, 21447424, 78191744, 285068032, 1039286528, 3788981248, 13813681664, 50361250816, 183604591616, 669376688128, 2440380925952, 8897021980672, 32436329517056, 118254790918144

Offset: 0

Author

Sean A. Irvine, Jun 07 2025

Keywords

• nonn
• easy
• walk

Comments

Number of walks of length n in the graph K_{1,1,1,2} starting at vertex 1 when the edges are given by {{0,1}, {0,3}, {0,4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}}.

Also, by symmetry, the number of walks of length n starting at vertex 3 (or 4) in the same graph.

Examples

			a(2)=14 because we have the walks 1-0-1, 1-0-3, 1-0-4, 1-2-1, 1-2-3, 1-2-4, 1-3-0, 1-3-1, 1-3-2, 1-3-4, 1-4-0, 1-4-1, 1-4-2, 1-4-3.
		

Links

• Sean A. Irvine, Walks on Graphs.

Crossrefs

Cf. A133592, A384711 (vertices 0, 2).

Programs

Maple

a:= n-> (<<0|1|0|1|1>, <1|0|1|1|1>, <0|1|0|1|1>, <1|1|1|0|1>, <1|1|1|1|0>>^n. <<1, 1, 1, 1, 1>>)[2, 1]:
seq(a(n), n=0..32);

Mathematica

CoefficientList[Series[(1+2*x) / (1-2*x-6*x^2), {x, 0, 32}], x]

Formula

a(n) = A133592(n+1)/2.