A054880 a(n) = 3*(9^n - 1)/4.

0, 6, 60, 546, 4920, 44286, 398580, 3587226, 32285040, 290565366, 2615088300, 23535794706, 211822152360, 1906399371246, 17157594341220, 154418349070986, 1389765141638880, 12507886274749926, 112570976472749340, 1013138788254744066, 9118249094292696600, 82064241848634269406, 738578176637708424660

Offset: 0

Author

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

Keywords

• nonn
• easy
• walk
• changed

Comments

Number of walks of length 2n+1 along the edges of a (3 dimensional) cube between two opposite vertices.

Urn A initially contains 3 labeled balls while urn B is empty. A ball is randomly selected and switched from one urn to the other. a(n)/3^(2n+1) is the probability that urn A is empty after 2n+1 switches. - Geoffrey Critzer, May 23 2013

Links

• G. C. Greubel, 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. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 5.
• Index entries for linear recurrences with constant coefficients, signature (10,-9).

Crossrefs

Cf. A002452, A015518, A046717, A066443.

Programs

GAP

List([0..30], n-> 3*(9^n -1)/4); # G. C. Greubel, Jul 14 2019

Magma

[3*(9^n -1)/4: n in [0..30]]; // G. C. Greubel, Jul 14 2019

Mathematica

Table[(2 n + 1)! Coefficient[Series[Sinh[x]^3, {x, 0, 2 n + 1}],
x^(2 n + 1)], {n, 0, 30}]  (* Geoffrey Critzer, May 23 2013 *)
LinearRecurrence[{10,-9},{0,6},30] (* Harvey P. Dale, Sep 17 2024 *)

PARI

vector(30, n, n--; 3*(9^n -1)/4) \\ G. C. Greubel, Jul 14 2019

Sage

[3*(9^n -1)/4 for n in (0..30)] # G. C. Greubel, Jul 14 2019

Formula

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

a(n) = 6*A002452(n).

sin(x)^3 = Sum_{k>=0} (-1)^(k+1)*a(k)*x^(2k+1)/(2k+1)!. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001

a(n) = A015518(2n+1) - 1 = (A046717(2n+1) - 1)/2. - M. F. Hasler, Mar 20 2008

a(n) = 9*a(n-1) + 6 with n > 0, a(0) = 0. - Vincenzo Librandi, Aug 07 2010

a(n) = A066443(n) - 1. - Georg Fischer, Nov 25 2018

E.g.f.: 3*(exp(9*x) - exp(x))/4. - G. C. Greubel, Jul 14 2019

a(n) = 10*a(n-1) - 9*a(n-2) with a(0) = 0 and a(1) = 6. - Miquel A. Fiol, Mar 09 2024