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

Author

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

Keywords

• nonn
• walk
• easy

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).

Crossrefs

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

Programs

Magma

[(4^n+(-1)^n*2^(n+1)+3*0^n)/6: n in [0..30]]; // Vincenzo Librandi, Apr 23 2015

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 *)

SageMath

[(4^n + (-1)^n*2^(n+1) + 3*0^n)/6 for n in range(31)] # G. C. Greubel, Feb 06 2023

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