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A024175 Expansion of g.f. (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, 18344191078400, 62630938517504 (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
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
Santiago Rojas-Rojas, Camila Muñoz, Edgar Barriga, Pablo Solano, Aldo Delgado, and Carla Hermann-Avigliano, Analytic Evolution for Complex Coupled Tight-Binding Models: Applications to Quantum Light Manipulation, arXiv:2310.12366 [quant-ph], 2023. See p. 12.
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
E.g.f.: (1 + exp(2*x)*(1 + 2*cosh(sqrt(2)*x) - sqrt(2)*sinh(sqrt(2)*x)))/4. - Stefano Spezia, Jun 14 2023
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. A211216.
Sequence in context: A162746 A148329 A293499 * A152226 A054393 A261589
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified May 22 06:48 EDT 2024. Contains 372743 sequences. (Running on oeis4.)