A122951 Number of walks from (0,0) to (n,n) in the region x >= y with the steps (1,0), (0,1), (2,0) and (0,2).

1, 1, 5, 22, 117, 654, 3843, 23323, 145172, 921508, 5942737, 38825546, 256431172, 1709356836, 11485249995, 77703736926, 528893901963, 3619228605738, 24884558358426, 171828674445330, 1191050708958096, 8284698825305832

Offset: 0

Author

Arvind Ayyer, Oct 25 2006

Keywords

• nonn
• nice
• walk

Comments

When this walk is further restricted to the subset of the plane x-y <= 2, this gives the sequence A046717. Similarly, the sequence for such a walk restricted to x-y <= w (w > 2) is not present in the OEIS. The reference provided proves recurrences for generating functions in w.

Examples

			a(2) = 5 because we can reach (2,2) in the following ways:
  (0,0),(1,0),(1,1),(2,1),(2,2)
  (0,0),(2,0),(2,2)
  (0,0),(1,0),(2,0),(2,2)
  (0,0),(2,0),(2,1),(2,2)
  (0,0),(1,0),(2,0),(2,1),(2,2).
		

Links

• Robert Israel, Table of n, a(n) for n = 0..1000
• Arvind Ayyer and Doron Zeilberger, The Number of [Old-Time] Basketball games with Final Score n:n where the Home Team was never losing but also never ahead by more than w Points, arXiv:math/0610734 [math.CO], 2006-2007.

Crossrefs

Cf. A000108, A046717.

Programs

Maple

N:= 100: # to get a[0] to a[N]
S:= series(RootOf(z^4*F^4-2*z^3*F^3-z^2*F^3+2*z^2*F^2+3*z*F^2-2*z*F-F+1,F), z, N+1):
seq(coeff(S,z,j),j=0..N); # Robert Israel, Feb 18 2013

Mathematica

f[x_] = (2x+Sqrt[4(x-2)x+1] - Sqrt[2]Sqrt[2x(-2x + Sqrt[4(x-2)x+1]-1) + Sqrt[4(x-2)x+1]+1]+1)/(4x^2);
CoefficientList[Series[f[x],{x,0,21}],x]
(* Jean-François Alcover, May 19 2011, after g.f. *)

Formula

In Maple, GF is given by solve(z^4*F^4 -2*z^3*F^3 -z^2*F^3 +2*z^2*F^2 +3*z*F^2 -2*z*F-F+1, F);