A094555 Number of walks of length n between two vertices on the same triangular face of a truncated tetrahedron (triangular prism).

0, 1, 1, 6, 11, 46, 111, 386, 1051, 3366, 9671, 29866, 87891, 267086, 794431, 2396946, 7163531, 21545206, 64526391, 193797626, 580955971, 1743741726, 5229477551, 15691927906, 47068793211, 141220360646, 423633119911, 1270955283786

Offset

0,4

Comments

Average of binomial and inverse binomial transforms of the Jacobsthal numbers A001045. - Paul Barry, Jan 04 2005

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 3.

Index entries for linear recurrences with constant coefficients, signature (2,5,-6).

Formula

G.f.: x*(1 - x - x^2)/((1 - x)*(1 + 2*x)*(1 - 3*x)).

a(n) = 3^n/6 - (-2)^n/6 + 1/6 - 0^n/6.

a(n) = 2*a(n-1) + 5*a(n-2) - 6*a(n-3) for n >= 4.

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*A001045(n-2k). - Paul Barry, Jan 04 2005

E.g.f.: exp(-2*x)*(exp(5*x) + exp(3*x) - exp(2*x) - 1)/6. - Stefano Spezia, Dec 26 2021

Mathematica

LinearRecurrence[{2, 5, -6}, {0, 1, 1, 6}, 30] (* Greg Dresden, Jun 19 2021 *)

Prog

(PARI) a(n) = if(n==0, 0, (3^n - (-2)^n + 1)/6) \\ Andrew Howroyd, Jun 15 2021

Crossrefs

Cf. A001045, A094554, A094556.

Sequence in context: A270727 A216269 A242488 * A271056 A271255 A219878

Adjacent sequences: A094552 A094553 A094554 * A094556 A094557 A094558

Keyword

easy,nonn

Author

Paul Barry, May 11 2004

Status

approved