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 A054879 Closed walks of length 2n along the edges of a cube based at a vertex. 15
 1, 3, 21, 183, 1641, 14763, 132861, 1195743, 10761681, 96855123, 871696101, 7845264903, 70607384121, 635466457083, 5719198113741, 51472783023663, 463255047212961, 4169295424916643, 37523658824249781, 337712929418248023, 3039416364764232201, 27354747282878089803, 246192725545902808221 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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 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). 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 MATHEMATICA nn = 40; Select[Range[0, nn]! CoefficientList[Series[Cosh[x]^3, {x, 0, nn}], x], # > 0 &]  (* Geoffrey Critzer, Dec 20 2012 *) PROG (MAGMA) [(3^(2*n)+3)/4: n in [0..25]]; // Vincenzo Librandi, Jun 30 2011 CROSSREFS Cf. A081294, A092812, A121822, A122983. Sequence in context: A295541 A192946 A216171 * A333090 A131763 A006199 Adjacent sequences:  A054876 A054877 A054878 * A054880 A054881 A054882 KEYWORD nonn,easy,walk AUTHOR Paolo Dominici (pl.dm(AT)libero.it), May 23 2000 STATUS approved

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Last modified May 9 03:39 EDT 2021. Contains 343685 sequences. (Running on oeis4.)