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 A177792 Number of paths from (0,0) to (n,n) avoiding 4 or more consecutive east steps and 4 or more consecutive north steps. 1
 1, 2, 6, 20, 62, 194, 616, 1972, 6344, 20498, 66486, 216352, 705982, 2309246, 7569420, 24857864, 81768144, 269369282, 888569354, 2934666604, 9702925752, 32113058042, 106379839060, 352698604852, 1170271492014, 3885821473458, 12911299418962, 42926732404728 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of binary strings of length 2*n containing n zeros and n ones, avoiding the patterns 0000 and 1111, see example. [Joerg Arndt, Dec 02 2013] LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 FORMULA a(n) = 2*Sum_{i=0..floor(n/3)} Sum_{j=0..floor((n-3*i)/2)} C(n-2*i-j,i) * C(n-3*i-j,j) * (Sum_{s=0..min(floor(n/3), floor((2*i+j)/2))} C(n-2*i-j,s) * C(n-2*i-j-s, 2*i+j-2*s) + Sum_{s=0..min(floor(n/3), floor((2*i+j+1)/2))} C(n-2*i-j-1,s) * C(n-2*i-j-s-1, 2*i+j+1-2*s)) if n>0; a(0) = 1. a(n) = [x^n y^n] (1+x+x^2+x^3)*(1+y+y^2+y^3) / (1-x*y-x*y^2 -x*y^3-x^2*y -x^2*y^2-x^2*y^3 -x^3*y-x^3*y^2 -x^3*y^3). a(n) ~ c * d^n / sqrt(Pi*n), where d = 1 + 1/3*(54-6*sqrt(33))^(1/3) + (2*(9+sqrt(33)))^(1/3) / 3^(2/3) = 3.382975767906237494122708536455034586... is the root of the equation 1 + d + 3*d^2 - d^3 = 0, and c = 2.106003170801818641958056379397216... is the root of the equation -4 - 80*c^2 - 616*c^4 + 143*c^6 = 0. - Vaclav Kotesovec, Aug 22 2014 EXAMPLE From Joerg Arndt, Dec 02 2013: (Start) The a(3) = 20 binary strings of length 3 containing 3 zeros and 3 ones, avoiding the patterns 0000 and 1111 are (putting dots for zeros) 01:  [ . . . 1 1 1 ] 02:  [ . . 1 . 1 1 ] 03:  [ . . 1 1 . 1 ] 04:  [ . . 1 1 1 . ] 05:  [ . 1 . . 1 1 ] 06:  [ . 1 . 1 . 1 ] 07:  [ . 1 . 1 1 . ] 08:  [ . 1 1 . . 1 ] 09:  [ . 1 1 . 1 . ] 10:  [ . 1 1 1 . . ] 11:  [ 1 . . . 1 1 ] 12:  [ 1 . . 1 . 1 ] 13:  [ 1 . . 1 1 . ] 14:  [ 1 . 1 . . 1 ] 15:  [ 1 . 1 . 1 . ] 16:  [ 1 . 1 1 . . ] 17:  [ 1 1 . . . 1 ] 18:  [ 1 1 . . 1 . ] 19:  [ 1 1 . 1 . . ] 20:  [ 1 1 1 . . . ] (End) MAPLE A177792a := proc(n, i, j, s, l) binomial(n-2*i-j, i)*binomial(n-3*i-j, j)*binomial(n-2*i-j-l, s) *binomial(n-2*i-j-l-s, 2*i+j-2*s+l) ; end proc: A177792 := proc(n) local a, i, j, slim, s ; if n=0 then return(1) fi; a := 0 ; for i from 0 to n/3 do for j from 0 to (n-3*i)/2 do slim := min( n/3, i+j/2) ; a := a+add( A177792a(n, i, j, s, 0), s=0..slim) ; slim := min( n/3, i+(j+1)/2) ; a := a+add( A177792a(n, i, j, s, 1), s=0..slim) ; end do: end do: 2*a; end proc: seq(A177792(n), n=0..16) ; # R. J. Mathar, May 31 2010 # second Maple Program: b:= proc(i, j, k) option remember; `if`(i<0 or j<0, 0,       `if`(i=0 and j=0, 1, `if`(k<3, b(i-1, j, max(k, 0)+1), 0)+       `if`(k>-3, 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<3, b[i-1, j, Max[k, 0]+1], 0] + If[k > -3, b[i, j-1, Min[k, 0]-1], 0]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 29 2017, after Alois P. Heinz *) CROSSREFS Sequence in context: A260696 A052958 A247076 * A193235 A199102 A053730 Adjacent sequences:  A177789 A177790 A177791 * A177793 A177794 A177795 KEYWORD nonn,walk AUTHOR Shanzhen Gao, May 13 2010 EXTENSIONS More terms from R. J. Mathar, May 31 2010 Edited by Alois P. Heinz, Jun 04 2011 STATUS approved

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Last modified May 16 19:46 EDT 2021. Contains 343951 sequences. (Running on oeis4.)