A384671 Expansion of (1-x^2) / (1-2*x-5*x^2+2*x^3).

1, 2, 8, 24, 84, 272, 916, 3024, 10084, 33456, 111284, 369680, 1228868, 4083568, 13572116, 45104336, 149902116, 498181680, 1655665268, 5502434704, 18286832388, 60774507760, 201978308052, 671255490128, 2230853504996, 7414027844528, 24639812233780

Offset

0,2

Comments

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

1---2

/|\ /|

0 | X |

\|/ \|

4---3.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Sean A. Irvine, Walks on Graphs.

Index entries for linear recurrences with constant coefficients, signature (2,5,-2).

Formula

a(n) = A384673(n)-A384673(n-1). - R. J. Mathar, Jun 07 2025

Example

a(2)=8 because we have the walks 0-1-0, 0-1-2, 0-1-3, 0-1-4, 0-4-0, 0-4-1, 0-4-2, 0-4-3.

Maple

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

Mathematica

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

Prog

(Magma) m:=35; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x^2) / (1-2*x-5*x^2+2*x^3))); // Vincenzo Librandi, Oct 10 2025

Crossrefs

Cf. A384672 (vertices 1 and 4), A384673 (vertices 2 and 3), A384646 (missing edge {2,4}).

Sequence in context: A034741 A063727 A085449 * A127362 A133443 A094038

Adjacent sequences: A384668 A384669 A384670 * A384672 A384673 A384674

Keyword

nonn,easy,walk

Author

Sean A. Irvine, Jun 05 2025

Status

approved