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%I #81 Oct 25 2023 09:31:06
%S 1,1,2,5,14,42,132,428,1416,4744,16016,54320,184736,629280,2145600,
%T 7319744,24979584,85262464,291057920,993641216,3392317952,11581727232,
%U 39541748736,135002491904,460924372992,1573688313856,5372896120832,18344191078400,62630938517504
%N Expansion of g.f. (x^3 - 6*x^2 + 5*x - 1)/((2*x - 1)*(2*x^2 - 4*x + 1)).
%C 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
%C 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
%H Vincenzo Librandi, <a href="/A024175/b024175.txt">Table of n, a(n) for n = 0..1000</a>
%H Giulio Cerbai, Anders Claesson, and Luca Ferrari, <a href="https://arxiv.org/abs/1907.08142">Stack sorting with restricted stacks</a>, arXiv:1907.08142 [cs.DS], 2019.
%H Michael Dairyko, Lara Pudwell, Samantha Tyner, and Casey Wynn, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p22">Non-contiguous pattern avoidance in binary trees</a>. Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227.
%H S. Felsner and D. Heldt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Felsner/felsner2.html">Lattice Path Enumeration and Toeplitz Matrices</a>, J. Int. Seq. 18 (2015) # 15.1.3.
%H Daniel Heldt, <a href="https://depositonce.tu-berlin.de/handle/11303/5553">On the mixing time of the face flip-and up/down Markov chain for some families of graphs</a>, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
%H M. Hyatt and J. Remmel, <a href="http://arxiv.org/abs/1208.1052">The classification of 231-avoiding permutations by descents and maximum drop</a>, arXiv preprint arXiv:1208.1052, 2012. - From _N. J. A. Sloane_, Dec 24 2012
%H Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, <a href="http://arxiv.org/abs/1201.6243">Marked mesh patterns in 132-avoiding permutations I,</a> arXiv:1201.6243v1 [math.CO], 2012 (Corollary 3, case k=6, pages 10-11). - From _N. J. A. Sloane_, May 09 2012
%H Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, <a href="http://www.emis.de/journals/INTEGERS/papers/p16/p16.Abstract.html">Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II</a>, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (<a href="http://arxiv.org/abs/1302.2274">arXiv:1302.2274</a>)
%H Huyile Liang, Jeffrey Remmel, and Sainan Zheng, <a href="https://arxiv.org/abs/1710.05795">Stieltjes moment sequences of polynomials</a>, arXiv:1710.05795 [math.CO], 2017, see page 13.
%H D. Necas and I. Ohlidal, <a href="https://doi.org/10.1364/OE.22.004499">Consolidated series for efficient calculation of the reflection and transmission in rough multilayers</a>, Optics Express, Vol. 22, 2014, No. 4; DOI:10.1364/OE.22.004499.
%H László Németh and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Nemeth/nemeth8.html">Sequences Involving Square Zig-Zag Shapes</a>, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
%H L. Pudwell, <a href="http://faculty.valpo.edu/lpudwell/slides/notredame.pdf">Pattern avoidance in trees</a> (slides from a talk, mentions many sequences), 2012. - From N. J. A. Sloane, Jan 03 2013
%H Santiago Rojas-Rojas, Camila Muñoz, Edgar Barriga, Pablo Solano, Aldo Delgado, and Carla Hermann-Avigliano, <a href="https://arxiv.org/abs/2310.12366">Analytic Evolution for Complex Coupled Tight-Binding Models: Applications to Quantum Light Manipulation</a>, arXiv:2310.12366 [quant-ph], 2023. See p. 12.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-10,4).
%F From _Herbert Kociemba_, Jun 11 2004: (Start)
%F a(n) = (1/4)*Sum_{r=1..7} sin(r*Pi/8)^2*(2*cos(r*Pi/8))^(2n), n >= 1.
%F a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3), n >= 4. (End)
%F a(n) = (1/4)*((2 + sqrt(2))^(n - 1) + (2 - sqrt(2))^(n - 1) + 2^n) for n >= 1. [_Richard Choulet_, Apr 19 2010]
%F a(n) = 2^(n - 2) + A006012(n-1)/2, n > 0. - R. J. Mathar, Mar 14 2011
%F G.f.: 1 / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x)))))). - _Michael Somos_, May 12 2012
%F 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
%e 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 428*x^7 + ...
%p 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
%t 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 *)
%o (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 */
%Y Cf. A006012, A030436 and A094803.
%Y Cf. A211216.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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