A384604 Expansion of (1-x^2) / (1-x-4*x^2+2*x^3).

1, 1, 4, 6, 20, 36, 104, 208, 552, 1176, 2968, 6568, 16088, 36424, 87640, 201160, 478872, 1108232, 2621400, 6096584, 14365720, 33509256, 78778968, 184084552, 432181912, 1010962184, 2371520728, 5551005640, 13015164184, 30476145288, 71434790744, 167309043528

Offset: 0

Author

Sean A. Irvine, Jun 04 2025

Keywords

• nonn
• easy
• walk

Comments

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

0 2

\ /|

1 |

/ \|

4 3.

Also, for n >= 1, the number of walks of length n-1 starting at vertex 1 in the same graph.

Examples

			a(3)=6 because we have the walks 0-1-0-1, 0-1-2-1, 0-1-2-3, 0-1-3-1, 0-1-3-2, 0-1-4-1.
		

Links

• Sean A. Irvine, Walks on Graphs.
• Index entries for linear recurrences with constant coefficients, signature (1,4,-2).

Crossrefs

Cf. A384605 (vertex 2), A213173 (missing edge {2,3}), A382683 (missing edge {1,4}).

Programs

Maple

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

Mathematica

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