This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A024175 Expansion of (x^3 - 6*x^2 + 5*x - 1)/((2*x - 1)*(2*x^2 - 4*x + 1)) 10
 1, 1, 2, 5, 14, 42, 132, 428, 1416, 4744, 16016, 54320, 184736, 629280, 2145600, 7319744, 24979584, 85262464, 291057920, 993641216, 3392317952, 11581727232, 39541748736, 135002491904, 460924372992, 1573688313856, 5372896120832 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of (s(0), s(1), ..., s(2*n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1, 2, ...., 2*n, s(0) = 1, s(2*n) = 1. - Herbert Kociemba, Jun 11 2004 Counts all paths of length (2*n), n >= 0, starting and ending at the initial node on the path graph P_7, see the Maple program. - Johannes W. Meijer, May 29 2010 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Giulio Cerbai, Anders Claesson, Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019. Michael Dairyko, Lara Pudwell, Samantha Tyner, Casey Wynn, Non-contiguous pattern avoidance in binary trees. Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227. S. Felsner, D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3. Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016. M. Hyatt and J. Remmel, The classification of 231-avoiding permutations by descents and maximum drop, arXiv preprint arXiv:1208.1052, 2012. - From N. J. A. Sloane, Dec 24 2012 Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (Corollary 3, case k=6, pages 10-11). - From N. J. A. Sloane, May 09 2012 Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274) Huyile Liang, Jeffrey Remmel, Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 13. D. Necas, I. Ohlidal, Consolidated series for efficient calculation of the reflection and transmission in rough multilayers, Optics Express, Vol. 22, 2014, No. 4; DOI:10.1364/OE.22.004499. L. Pudwell, Pattern avoidance in trees (slides from a talk, mentions many sequences), 2012. - From N. J. A. Sloane, Jan 03 2013 Index entries for linear recurrences with constant coefficients, signature (6,-10,4). FORMULA From Herbert Kociemba, Jun 11 2004: (Start) a(n) = (1/4)*Sum_{r=1..7} sin(r*Pi/8)^2*(2*cos(r*Pi/8))^(2n), n >= 1. a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3), n >= 4. (End) a(n) = (1/4)*((2 + sqrt(2))^(n - 1) + (2 - sqrt(2))^(n - 1) + 2^n) for n >= 1. [Richard Choulet, Apr 19 2010] a(n) = 2^(n - 2) + A006012(n-1)/2, n > 0. - R. J. Mathar, Mar 14 2011 G.f.: 1 / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x)))))). - Michael Somos, May 12 2012 EXAMPLE 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 428*x^7 + ... MAPLE with(GraphTheory): G:=PathGraph(7): A:= AdjacencyMatrix(G): nmax:=26; n2:=2*nmax: for n from 0 to n2 do B(n):=A^n; a(n):=B(n)[1, 1]; od: seq(a(2*n), n=0..nmax); # Johannes W. Meijer, May 29 2010 MATHEMATICA CoefficientList[Series[(x^3-6*x^2+5*x-1)/((2*x-1)*(2*x^2-4*x+1)), {x, 0, 30}], x] (* Vincenzo Librandi, May 10 2012 *) PROG (PARI) {a(n) = local(A); A = 1; for( i=1, 6, A = 1 / (1 - x*A)); polcoeff( A + x * O(x^n), n)} /* Michael Somos, May 12 2012 */ CROSSREFS Cf. A006012, A030436 and A094803. Cf. A211216. Sequence in context: A162746 A148329 A293499 * A152226 A054393 A261589 Adjacent sequences:  A024172 A024173 A024174 * A024176 A024177 A024178 KEYWORD nonn,easy AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 16 06:30 EDT 2019. Contains 328050 sequences. (Running on oeis4.)