A384599 Expansion of (1+3*x+2*x^2) / (1-4*x^2-2*x^3).

1, 3, 6, 14, 30, 68, 148, 332, 728, 1624, 3576, 7952, 17552, 38960, 86112, 190944, 422368, 936000, 2071360, 4588736, 10157440, 22497664, 49807232, 110305536, 244224256, 540836608, 1197508096, 2651794944, 5871705600, 13002195968, 28790412288, 63752195072

Offset

0,2

Comments

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

3

/|

0-1-2 |

\|

4.

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 (0,4,2).

Example

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

Maple

a:= n-> (<<0|1|0>, <0|0|1>, <2|4|0>>^n. <<1, 3, 6>>)[1, 1]:
seq(a(n), n=0..31);  # Alois P. Heinz, Jun 04 2025

Mathematica

CoefficientList[Series[(1 + 3*x + 2*x^2)/(1 - 4*x^2 - 2*x^3), {x, 0, 31}], x] (* Michael De Vlieger, Jun 04 2025 *)
LinearRecurrence[{0, 4, 2}, {1, 3, 6}, 33] (* Vincenzo Librandi, Oct 13 2025 *)

Prog

(Magma) I:=[1, 3, 6]; [n le 3 select I[n] else 4*Self(n-2)+2*Self(n-3): n in [1..35]]; // Vincenzo Librandi, Oct 13 2025

Crossrefs

Cf. A384598 (vertices 0 and 1), A384600 (vertex 3), A062112 (missing edge {3,4}), A382683 (missing edge {0,1}).

Sequence in context: A196450 A131244 A077926 * A091601 A339154 A063119

Adjacent sequences: A384596 A384597 A384598 * A384600 A384601 A384602

Keyword

nonn,walk,easy

Author

Sean A. Irvine, Jun 04 2025

Status

approved