A054881 Number of walks of length n along the edges of an octahedron starting and ending at a vertex and also ( with a(0)=0 ) between two opposite vertices.

1, 0, 4, 8, 48, 160, 704, 2688, 11008, 43520, 175104, 698368, 2797568, 11182080, 44744704, 178946048, 715849728, 2863267840, 11453333504, 45812809728, 183252287488, 733007052800, 2932032405504, 11728121233408

Offset

0,3

Links

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

R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 7.

Index entries for linear recurrences with constant coefficients, signature (2,8).

Formula

a(n) = 4*A003683(n-1) + 0^n/2, n >= 0.

a(n) = (4^n + (-1)^n*2^(n+1) + 3*0^n)/6.

G.f.: (1/6)*(3 + 2/(1+2*x) + 1/(1-4*x)).

From L. Edson Jeffery, Apr 22 2015: (Start)

G.f.: (1-2*x-4*x^2)/((1+2*x)*(1-4*x)).

a(n) = 8*A246036(n-3) + 0^n/2, n >= 0. (End)

a(n) = 2^n*A001045(n-1) + (1/2)*[n=0] = 2^n*(2^(n-1) + (-1)^n)/3 + (1/2)*[n=0], n >= 0. - Ralf Steiner, Aug 27 2020, edited by M. F. Hasler, Sep 11 2020

E.g.f.: (1/6)*(exp(4*x) + 2*exp(-2*x) + 3). - G. C. Greubel, Feb 06 2023

Mathematica

CoefficientList[Series[(1-2*x-4*x^2)/((1+2x)*(1-4x)), {x, 0, 40}], x] (* L. Edson Jeffery, Apr 22 2015 *)
LinearRecurrence[{2, 8}, {1, 0, 4}, 41] (* G. C. Greubel, Feb 06 2023 *)

Prog

(Magma) [(4^n+(-1)^n*2^(n+1)+3*0^n)/6: n in [0..30]]; // Vincenzo Librandi, Apr 23 2015
(SageMath) [(4^n + (-1)^n*2^(n+1) + 3*0^n)/6 for n in range(31)] # G. C. Greubel, Feb 06 2023

Crossrefs

Cf. A001045, A003683, A054882, A054883, A054884, A054885, A246036.

Sequence in context: A087261 A249572 A078236 * A045882 A051681 A267987

Adjacent sequences: A054878 A054879 A054880 * A054882 A054883 A054884

Keyword

nonn,walk,easy

Author

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Status

approved