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 A177790 Number of paths from (0,0) to (n,n) avoiding 3 or more consecutive east steps and 3 or more consecutive north steps. 3
 1, 2, 6, 14, 34, 84, 208, 518, 1296, 3254, 8196, 20700, 52404, 132942, 337878, 860142, 2192902, 5598144, 14308378, 36610970, 93770358, 240390602, 616787116, 1583765724, 4069672444, 10464501074, 26924530158, 69315481778, 178545361842, 460138256784 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) equals the number of different permutations of n 0's and n 1's such that no more than two occurrences of the same number ever appear in a row. - Dave R.M. Langers, Apr 07 2016 This also equals the number of possible different rows or columns that may occur in a 2n-by-2n binary puzzle. - Dave R.M. Langers, Apr 07 2016 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..750 Daiki Miyahara, Léo Robert, Pascal Lafourcade, So Takeshige, Takaaki Mizuki, Kazumasa Shinagawa, Atsuki Nagao, Hideaki Sone, Card-Based ZKP Protocols for Takuzu and Juosan, 10th International Conference on Fun with Algorithms (FUN 2021). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 20:1-21. FORMULA a(n) = Sum_{i=0..floor(n/2)} 2*C(n-i,i)^2 + C(n-i,i)*C(n-i-1,i+1) + C(n-i,i)*C(n-i+1,i-1). a(n) = [x^n y^n] (-(1+x+x^2)*(1+y+y^2) / (-1+x*y+x*y^2+x^2*y+x^2*y^2)). G.f.: 1 + ((1-t)^2*sqrt((1+t+t^2)*(1-3*t+t^2))-(1-3*t+t^2)*(1+t^2)) / (t^2*(1-3*t+t^2)). Recurrence: (n-2)*(n-1)*(n+2)*a(n) = 2*(n-2)*n*(n+1)*a(n-1) + (n-1)*(n^2 - 2*n - 4)*a(n-2) + 2*(n-3)*(n-2)*n*a(n-3) - (n-4)*(n-1)*n*a(n-4). - Vaclav Kotesovec, Aug 18 2013 a(n) ~ (3+sqrt(5))^n * sqrt((15+7*sqrt(5))/(5*Pi*n))/2^(n-1/2). - Vaclav Kotesovec, Aug 18 2013 EXAMPLE For n=3, the a(3)=14 possible arrangements are 001011, 001101, 010011, 010101, 010110, 011001, 011010, 100101, 100110, 101001, 101010, 101100, 110010, and 110100. - Dave R.M. Langers, Apr 07 2016 MAPLE b:= proc(i, j, k) option remember; `if`(i<0 or j<0, 0,       `if`(i=0 and j=0, 1, `if`(k<2, b(i-1, j, max(k, 0)+1), 0)+       `if`(k>-2, b(i, j-1, min(k, 0)-1), 0)))     end: a:= n-> b(n, n, 0): seq(a(n), n=0..30);  # Alois P. Heinz, Jun 01 2011 MATHEMATICA b[i_, j_, k_] := b[i, j, k] = If[i<0 || j<0, 0, If[i == 0 && j == 0, 1, If[k<2, b[i-1, j, Max[k, 0]+1], 0] + If[k > -2, b[i, j-1, Min[k, 0] - 1], 0]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *) CROSSREFS Equals twice A003440 (number of binary vectors with restricted repetitions). Sequence in context: A182644 A099425 A186523 * A307068 A269506 A292816 Adjacent sequences:  A177787 A177788 A177789 * A177791 A177792 A177793 KEYWORD nonn,walk AUTHOR Shanzhen Gao, May 13 2010 EXTENSIONS Edited by Alois P. Heinz, Jun 03 2011 STATUS approved

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Last modified June 19 07:26 EDT 2021. Contains 345126 sequences. (Running on oeis4.)