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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). 8
1, 1, 5, 22, 117, 654, 3843, 23323, 145172, 921508, 5942737, 38825546, 256431172, 1709356836, 11485249995, 77703736926, 528893901963, 3619228605738, 24884558358426, 171828674445330, 1191050708958096, 8284698825305832 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A127618 A127619 A127620 * A331836 A184181 A020003
KEYWORD
nonn,nice,walk
AUTHOR
Arvind Ayyer, Oct 25 2006
STATUS
approved

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Last modified June 16 02:19 EDT 2024. Contains 373416 sequences. (Running on oeis4.)