A024175 Expansion of (x^3 - 6*x^2 + 5*x - 1)/((2*x - 1)*(2*x^2 - 4*x + 1))

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

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, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.

Michael Dairyko, Lara Pudwell, Samantha Tyner, and Casey Wynn, Non-contiguous pattern avoidance in binary trees. Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227.

S. Felsner and 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, and 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, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 13.

D. Necas and 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.

László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.

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

N. J. A. Sloane

Status

approved