cp's OEIS Frontend

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)}.

%I A122680 #30 Apr 24 2025 02:13:20
%S A122680 1,2,14,106,784,6040,47332,375196,3001966,24190148,196034522,
%T A122680 1596030740,13044459766,106961525744,879512777006,7249483605580,
%U A122680 59881171431050,495545064567260,4107666916668414,34099685718629264,283454909832384416,2359069189033880228
%N 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)}.
%C A122680 Diagonal of the rational function 1 / (1 - x - y - x^2 - y^2 - x^3 - y^3). - _Ilya Gutkovskiy_, Apr 23 2025
%H A122680 Alois P. Heinz, <a href="/A122680/b122680.txt">Table of n, a(n) for n = 0..1000</a>
%H A122680 Moa Apagodu and Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/appswz.html">FIVE Applications of Wilf-Zeilberger Theory to Enumeration and Probability</a>; <a href="/A036692/a036692.pdf">Local copy</a> [Pdf file only, no active links]
%H A122680 M. Apagodu and D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/tokhniot/AppsWZmulti">Maple package to generate recurrence</a>; <a href="/A122680/a122680.txt">Local copy</a>
%H A122680 M. Apagodu and D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/tokhniot/oAppsWZmulti4">6th order recurrence</a>; <a href="/A122680/a122680_1.txt">Local copy</a>
%e A122680 a(1) = 2 because the number of ways of getting the score to be (1,1) is
%e A122680 (0,0) -> (1,0) -> (1,1),
%e A122680 (0,0) -> (0,1) -> (1,1).
%p A122680 b:= proc(x, y) option remember; `if`(x<0 or y<0, 0,
%p A122680       `if`([x, y]=[0, 0], 1, add(b(x-i, y) +b(x, y-i), i=1..3)))
%p A122680     end:
%p A122680 a:= n-> b(n, n):
%p A122680 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jul 24 2011
%t A122680 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}]]];
%t A122680 a[n_] := b[n, n];
%t A122680 a /@ Range[0, 30] (* _Jean-François Alcover_, May 13 2020, after _Alois P. Heinz_ *)
%K A122680 nonn,walk
%O A122680 0,2
%A A122680 _Doron Zeilberger_, Oct 20 2006