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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
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
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
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