A122680 Number of possible basketball games that end in a tie (n:n). Equivalently, the number of walks on the square lattice from (0,0) to (n,n) where the allowed steps are {(1,0),(2,0),(3,0), (0,1),(0,2),(0,3)}.

1, 2, 14, 106, 784, 6040, 47332, 375196, 3001966, 24190148, 196034522, 1596030740, 13044459766, 106961525744, 879512777006, 7249483605580, 59881171431050, 495545064567260, 4107666916668414, 34099685718629264, 283454909832384416, 2359069189033880228

Offset: 0

Author

Doron Zeilberger, Oct 20 2006

Keywords

• nonn
• walk

Comments

Diagonal of the rational function 1 / (1 - x - y - x^2 - y^2 - x^3 - y^3). - Ilya Gutkovskiy, Apr 23 2025

Examples

			a(1) = 2 because the number of ways of getting the score to be (1,1) is
(0,0) -> (1,0) -> (1,1),
(0,0) -> (0,1) -> (1,1).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Moa Apagodu and Doron Zeilberger, FIVE Applications of Wilf-Zeilberger Theory to Enumeration and Probability; Local copy [Pdf file only, no active links]
• M. Apagodu and D. Zeilberger, Maple package to generate recurrence; Local copy
• M. Apagodu and D. Zeilberger, 6th order recurrence; Local copy

Programs

Maple

b:= proc(x, y) option remember; `if`(x<0 or y<0, 0,
      `if`([x, y]=[0, 0], 1, add(b(x-i, y) +b(x, y-i), i=1..3)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 24 2011

Mathematica

b[x_, y_] := b[x, y] = If[x < 0 || y < 0, 0, If[{x, y} == {0, 0}, 1, Sum[b[x - i, y] + b[x, y - i], {i, 3}]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, May 13 2020, after Alois P. Heinz *)