A192365 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(2,0),(0,1),(0,2),(1,1),(2,2).

1, 3, 22, 165, 1327, 10950, 92045, 783579, 6733966, 58294401, 507579829, 4440544722, 39000863629, 343677908223, 3037104558574, 26904952725061, 238854984979423, 2124492829796598, 18927927904130617, 168888613467092895, 1508973226894216106, 13498652154574126523, 120886709687492946083

Offset: 0

Author

Eric Werley, Jun 29 2011

Keywords

• nonn

Comments

Diagonal of the rational function 1 / (1 - x - y - x^2 - y^2 - x*y - (x*y)^2). - Ilya Gutkovskiy, Apr 23 2025

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Programs

Maple

p4 := x^4+6*x^3+7*x^2-10*x+1;
ogf := sqrt( ((2*x^2+6*x-3)/p4 - 2/sqrt(p4))/(4*x^2-4*x-5) );
series(ogf, x=0, 30);  # Mark van Hoeij, Apr 16 2013
# second Maple program:
b:= proc(x, y) option remember; `if`(min(x, y)<0, 0,
      `if`(max(x, y)=0, 1, add(b(x, y-j)+
         b(x-j, y)+b(x-j, y-j), j=1..2)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 16 2017

Mathematica

b[x_, y_] := b[x, y] = If[Min[x, y] < 0, 0, If[Max[x, y] == 0, 1, Sum[b[x, y - j] + b[x - j, y] + b[x - j, y - j], {j, 1, 2}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 23 2017, after Alois P. Heinz *)

PARI

/* same as in A092566 but use */
steps=[[0,1], [0,2], [1,0], [2,0], [1,1], [2,2]];
/* Joerg Arndt, Jun 30 2011 */

Formula

G.f.: sqrt( ((2*x^2+6*x-3)/p4 - 2/sqrt(p4))/(4*x^2-4*x-5) ) where p4 = x^4+6*x^3+7*x^2-10*x+1. - Mark van Hoeij, Apr 16 2013

Extensions

Terms > 507579829 from Joerg Arndt, Jun 30 2011