A384648 Expansion of (1+2*x+x^2) / (1-x-5*x^2-2*x^3).

1, 3, 9, 26, 77, 225, 662, 1941, 5701, 16730, 49117, 144169, 423214, 1242293, 3646701, 10704594, 31422685, 92239057, 270761670, 794802325, 2333088789, 6848623754, 20103672349, 59012968697, 173228577950, 508500766133, 1492669593277, 4381630579842

Offset: 0

Author

Sean A. Irvine, Jun 05 2025

Keywords

• nonn
• easy
• walk

Comments

Number of walks of length n starting at vertex 0 in the following graph:

1---2

/|\ |

0 | \ |

\| \|

4---3.

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

Examples

			a(2)=9 because we have the walks 3-1-0, 3-1-2, 3-1-3, 3-1-4, 3-2-1, 3-2-3, 3-4-0, 3-4-1, 3-4-3.
		

Links

• Sean A. Irvine, Walks on Graphs.
• Index entries for linear recurrences with constant coefficients, signature (1,5,2).

Crossrefs

Cf. A384646 (vertices 0 and 2), A384647 (vertex 1), A077937 (missing edge {1,3}).

Programs

Maple

a:= n->  (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <1|1|0|1|0>>^n. <<1,1,1,1,1>>)[4,1]:
seq(a(n), n=0..32);

Mathematica

CoefficientList[Series[(1+2*x+x^2) / (1-x-5*x^2-2*x^3), {x, 0, 32}], x]