A384642 Expansion of (1+x-x^2+x^3) / (1-x-5*x^2+x^3+2*x^4).

1, 2, 6, 16, 42, 112, 294, 780, 2054, 5436, 14338, 37904, 100050, 264360, 698030, 1843972, 4869662, 12862772, 33971050, 89727304, 236980458, 625920384, 1653153270, 4366320124, 11532205174, 30458811756, 80447210962, 212476424320, 561189257026, 1482206544152

Offset: 0

Author

Sean A. Irvine, Jun 05 2025

Keywords

• nonn
• easy
• walk

Comments

Number of walks of length n starting at vertex 3 in the following graph:

2

/|\

0-1 | 3

\|/

4.

Examples

			a(2)=6 because we have the walks 3-2-1, 3-2-3, 3-2-4, 3-4-1, 3-4-2, 3-4-3.
		

Links

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

Crossrefs

Cf. A384640 (vertices 0, 1), A384641 (vertex 2), A005824 (missing edge {2,4}), A026597 (missing edge {0,1}).

Programs

Maple

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

Mathematica

CoefficientList[Series[(1+x-x^2+x^3) / (1-x-5*x^2+x^3+2*x^4), {x, 0, 32}], x]