A384673 Expansion of (1+x) / (1-2*x-5*x^2+2*x^3).

1, 3, 11, 35, 119, 391, 1307, 4331, 14415, 47871, 159155, 528835, 1757703, 5841271, 19413387, 64517723, 214419839, 712601519, 2368266787, 7870701491, 26157533879, 86932041639, 288910349691, 960165839819, 3191019344815, 10605047189343, 35244859423123

Offset

0,2

Comments

Number of walks of length n starting at vertex 2 in the following graph:

1---2

/|\ /|

0 | X |

\|/ \|

4---3.

Also, by symmetry, the number of walks of length n starting at 3 in the same graph.

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).

Example

a(2)=11 because we have the walks 2-1-0, 2-1-2, 2-1-3, 2-1-4, 2-3-1, 2-3-2, 2-3-4, 2-4-0, 2-4-1, 2-4-2, 2-4-3.

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>>)[3, 1]:
seq(a(n), n=0..32);

Mathematica

CoefficientList[Series[(1+x) / (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) / (1-2*x-5*x^2+2*x^3))); // Vincenzo Librandi, Oct 10 2025

Crossrefs

Cf. A384671 (vertex 0), A384672 (vertices 1 and 4), A384646 (missing edge {2,4}).

Sequence in context: A320087 A014335 A383904 * A147474 A247417 A222286

Adjacent sequences: A384670 A384671 A384672 * A384674 A384675 A384676

Keyword

nonn,easy,walk

Author

Sean A. Irvine, Jun 05 2025

Status

approved