A384647 Expansion of (1+3*x+x^2) / (1-x-5*x^2-2*x^3).

1, 4, 10, 32, 90, 270, 784, 2314, 6774, 19912, 58410, 171518, 503392, 1477802, 4337798, 12733592, 37378186, 109721742, 322079856, 945444938, 2775287702, 8146672104, 23914000490, 70197936414, 206061283072, 604878966122, 1775581254310, 5212098651064

Offset: 0

Author

Sean A. Irvine, Jun 05 2025

Keywords

• nonn
• easy
• walk

Comments

Number of walks of length n starting at vertex 1 in the following graph:

1---2

/|\ |

0 | \ |

\| \|

4---3.

Examples

			a(2)=10 because we have the walks 1-0-1, 1-0-4, 1-2-1, 1-2-3, 1-3-1, 1-3-2, 1-3-4, 1-4-0, 1-4-1, 1-4-3.
		

Links

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

Crossrefs

Cf. A384646 (vertices 0, 2), A384648 (vertices 3 and 4), 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>>)[2,1]:
seq(a(n), n=0..32);

Mathematica

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