A057648 Number of excursions of length n on the upper-right part of the hexagonal lattice.

1, 0, 2, 2, 13, 34, 158, 594, 2665, 11558, 53320, 247488, 1181266, 5708884, 28049474, 139417402, 701063005, 3559326294, 18233244530, 94140532624, 489573775236, 2562613997512, 13493827469116, 71441865994904

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

Cf. A002898, A057647.

Sequence in context: A173466 A151367 A368957 * A282460 A327930 A068511

Adjacent sequences: A057645 A057646 A057647 * A057649 A057650 A057651

Keyword

nonn

Author

Cyril Banderier, Oct 12 2000

Extensions

Title corrected by Sean A. Irvine, Jun 22 2022

Status

approved