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

0,3

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.

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

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);

Example

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).

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. *)

Crossrefs

Cf. A000108, A046717.

Sequence in context: A127618 A127619 A127620 * A331836 A184181 A020003

Adjacent sequences: A122948 A122949 A122950 * A122952 A122953 A122954

Keyword

nonn,nice,walk

Author

Arvind Ayyer, Oct 25 2006

Status

approved