A054879 Closed walks of length 2n along the edges of a cube based at a vertex.

1, 3, 21, 183, 1641, 14763, 132861, 1195743, 10761681, 96855123, 871696101, 7845264903, 70607384121, 635466457083, 5719198113741, 51472783023663, 463255047212961, 4169295424916643, 37523658824249781, 337712929418248023, 3039416364764232201, 27354747282878089803, 246192725545902808221

Offset: 0

Author

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

Keywords

• nonn
• easy
• walk

Comments

a(n) is the number of words of length 2n on alphabet {0,1,2} with an even number (possibly zero) of each letter. - Geoffrey Critzer, Dec 20 2012

Equivalently, the cogrowth sequence of the 8-element group C2^3. - Sean A. Irvine, Nov 04 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• G. Benkart and D. Moon, A Schur-Weyl Duality Approach to Walking on Cubes, arXiv preprint arXiv:1409.8154 [math.RT], 2014 and Ann. Combin. 20 (3) (2016) 397-417
• R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.
• G. R. Franssens, On a number pyramid related to the binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
• Katarzyna Grygiel, Pawel M. Idziak and Marek Zaionc, How big is BCI fragment of BCK logic, arXiv preprint arXiv:1112.0643 [cs.LO], 2011. [From _N. J. A. Sloane_, Feb 21 2012]
• Ji-Hwan Jung, Oriented Riordan graphs and their fractal property, arXiv:2009.01677 [math.CO], 2020.
• R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 5.
• L. Reyzin, Number of closed walks on an n-cube, Mathoverflow.
• Index entries for linear recurrences with constant coefficients, signature (10,-9).

Crossrefs

Cf. A081294, A092812, A121822, A122983.

Programs

Magma

[(3^(2*n)+3)/4: n in [0..25]]; // Vincenzo Librandi, Jun 30 2011

Mathematica

nn = 40; Select[Range[0, nn]! CoefficientList[Series[Cosh[x]^3, {x, 0, nn}], x], # > 0 &]  (* Geoffrey Critzer, Dec 20 2012 *)
Table[(3^(2n)+3)/4,{n,0,30}] (* or *) LinearRecurrence[{10,-9},{1,3},30] (* Harvey P. Dale, Mar 17 2023 *)

Formula

a(n) = (3^(2*n)+3)/4.

G.f.: 1/4*1/(1-9*x)+3/4*1/(1-x).

a(n) = Sum_{k=0..n} 3^k*4^(n-k)*A121314(n,k). - Philippe Deléham, Aug 26 2006

E.g.f.: cosh^3(x). O.g.f.: 1/(1-3*1*x/(1-2*2*x/(1-1*3*x))) (continued fraction). - Peter Bala, Nov 13 2006

(-1)^n*a(n) = Sum_{k=0..n} A086872(n,k)*(-4)^(n-k). - Philippe Deléham, Aug 17 2007

a(n) = (1/2^3)*Sum_{j = 0..3} binomial(3,j)*(3 - 2*j)^(2*n). See Reyzin link. - Peter Bala, Jun 03 2019

a(n) = 9*a(n-1) - 6. - Klaus Purath, Mar 13 2021