OFFSET
0,3
COMMENTS
Excursions = walks from the origin to the origin.
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice. - Sean A. Irvine, Jun 22 2022
LINKS
Robert Israel, Table of n, a(n) for n = 0..1000
C. Banderier, Analytic combinatorics of random walks and planar maps, PhD Thesis, 2001.
FORMULA
G.f.: (1-2*x)*hypergeom([-1/2, 1/2],[2],16*x^2/(1-2*x)^2)/(4*x^2) - (2*x+1)*((1-6*x)*hypergeom([1/3, 2/3],[2],27*x^2*(2*x+1))+1/2)/(6*x^2). - Mark van Hoeij, Dec 08 2014
a(n) ~ (2*sqrt(3) - 3) * 2^n * 3^(n+2) / (Pi*n^3). - Vaclav Kotesovec, Apr 30 2024
MAPLE
gf:=(1-2*x)*hypergeom([-1/2, 1/2], [2], 16*x^2/(1-2*x)^2)/(4*x^2) - (2*x+1)*((1-6*x)*hypergeom([1/3, 2/3], [2], 27*x^2*(2*x+1))+1/2)/(6*x^2):
S:= series(gf, x, 103):
seq(coeff(S, x, j), j=0..100); # Robert Israel, Dec 08 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Cyril Banderier, Oct 12 2000
EXTENSIONS
Title corrected by Sean A. Irvine, Jun 22 2022
STATUS
approved