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, 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

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