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 A066443 Number of distinct paths of length 2n+1 along edges of a unit cube between two fixed adjacent vertices. 18
 1, 7, 61, 547, 4921, 44287, 398581, 3587227, 32285041, 290565367, 2615088301, 23535794707, 211822152361, 1906399371247, 17157594341221, 154418349070987, 1389765141638881, 12507886274749927, 112570976472749341 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS All members of sequence are also hex, or central hexagonal, numbers (A003215). (If n is a hex number, 9n - 2 is always a hex number; see recurrence.) - Matthew Vandermast, Mar 30 2003 The sequence 1,1,7,61,547,... with g.f. (1-9x+6x^2)/((1-x)(1-9x)) and a(n) = A054879(n)/3 + 2*0^n/3 gives the denominators in the probability that a random walk on the cube returns to its starting corner on the 2n-th step. - Paul Barry, Mar 11 2004 Equals row sums of even row terms of triangle A158303. - Gary W. Adamson, Mar 15 2009 It appears that a(n) is the n-th record value in A120437, which gives the  differences of A037314 (positive integers n such that the sum of the base 3 digits of n equals the sum of the base 9 digits of n). - John W. Layman, Dec 14 2010 Numbers in base 9 are 1, 6+1, 66+1, 666+1, 6666+1, 66666+1, etc.; that is, n 6's + 1. - Yuchun Ji, Aug 15 2019 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 G. Benkart, 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. E. Estrada and J. A. de la Pena, From Integer Sequences to Block Designs via Counting Walks in Graphs, arXiv preprint arXiv:1302.1176 [math.CO], 2013. - N. J. A. Sloane, Feb 28 2013 E. Estrada and J. A. de la Pena, Integer sequences from walks in graphs, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 3, 78-84. M. Kac, Random walk and the theory of Brownian motion, Amer. Math. Monthly, 54:369-391, 1947. R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 5. Vladimir Pletser, Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers, arXiv:1409.7969 [math.NT], 2014. Eric Weisstein's World of Mathematics, Repunit Index entries for linear recurrences with constant coefficients, signature (10,-9). FORMULA a(n) = (3^(2*n+1)+1)/4. - Vladeta Jovovic, Dec 22 2002 a(n) = 9*a(n-1) - 2. - Matthew Vandermast, Mar 30 2003 From Paul Barry, Apr 21 2003: (Start) G.f.: (1-3*x)/((1-x)*(1-9*x)). E.g.f.: (3*exp(9*x) + exp(x))/4. (End) a(n) = (-1)^n times the (i, i)-th element of M^n (for any i), where M = ((1, 1, 1, -2), (1, 1, -2, 1), (1, -2, 1, 1), (-2, 1, 1, 1)). - Simone Severini, Nov 25 2004 a(n) = Sum_{k=0..n} binomial(2*n+1, 2*k)*4^(n-k). - Paul Barry, Jan 22 2005 a(n) = A054880(n) + 1. a(n) = A057660(3^n). - Henry Bottomley, Nov 08 2015 a(n) = Sum_{k=0..2n} (-3)^k == 1 + Sum_{k=1..n} 2*3^(2k-1). - Bob Selcoe, Aug 21 2016 a(n) = 3^(2*n+1) * a(-1-n) for all n in Z. - Michael Somos, Jul 02 2017 a(n) = 6*A002452(n) + 1. - Yuchun Ji, Aug 15 2019 EXAMPLE From Michael B. Porter, Aug 22 2016: (Start) Give coordinates (a,b,c) to the vertices of the cube, where a, b, and c are either 0 or 1. For n = 1, the a(1) = 7 paths of length 2n + 1 = 3 from (0,0,0) to (0,0,1) are: (0,0,0) -> (0,0,1) -> (0,0,0) -> (0,0,1) (0,0,0) -> (0,0,1) -> (0,1,1) -> (0,0,1) (0,0,0) -> (0,0,1) -> (1,0,1) -> (0,0,1) (0,0,0) -> (0,1,0) -> (0,0,0) -> (0,0,1) (0,0,0) -> (0,1,0) -> (0,1,1) -> (0,0,1) (0,0,0) -> (1,0,0) -> (0,0,0) -> (0,0,1) (0,0,0) -> (1,0,0) -> (1,0,1) -> (0,0,1) (End) MAPLE seq((3^(2*n+1) + 1)/4, n=0..18); # Zerinvary Lajos, Jun 16 2007 MATHEMATICA NestList[9 # - 2 &, 1, 18] (* or *) Table[(3^(2 n + 1) + 1)/4, {n, 0, 18}] (* or *) CoefficientList[Series[(1 - 3 x)/((1 - x) (1 - 9 x)), {x, 0, 18}], x] (* Michael De Vlieger, Aug 22 2016 *) PROG (MAGMA) [(3^(2*n+1)+1)/4: n in [0..20]]; // Vincenzo Librandi, Jun 16 2011 (PARI) a(n)=3^(2*n+1)\/4 \\ Charles R Greathouse IV, Jul 02 2013 (PARI) Vec((1-3*x)/((1-x)*(1-9*x)) + O(x^50)) \\ Altug Alkan, Nov 13 2015 CROSSREFS Cf. A158303, A037314, A120437, A083234 (binomial transform), A083233 (inverse binomial transform), A054879 (recurrent walks), A125857 (walks ending on face diagonal), A054880 (walks ending on space diagonal). Sequence in context: A155599 A104093 A015572 * A108448 A218473 A098659 Adjacent sequences:  A066440 A066441 A066442 * A066444 A066445 A066446 KEYWORD nonn,easy AUTHOR John W. Layman, Aug 12 2002 EXTENSIONS Corrected by Vladeta Jovovic, Dec 22 2002 STATUS approved

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Last modified May 17 22:15 EDT 2021. Contains 343992 sequences. (Running on oeis4.)