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%I #20 Jun 22 2022 20:29:08
%S 1,0,2,2,13,34,158,594,2665,11558,53320,247488,1181266,5708884,
%T 28049474,139417402,701063005,3559326294,18233244530,94140532624,
%U 489573775236,2562613997512,13493827469116,71441865994904
%N Number of excursions of length n on the upper-right part of the hexagonal lattice.
%C Excursions = walks from the origin to the origin.
%C 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
%H Robert Israel, <a href="/A057648/b057648.txt">Table of n, a(n) for n = 0..1000</a>
%H C. Banderier, <a href="http://algo.inria.fr/banderier/">Analytic combinatorics of random walks and planar maps</a>, PhD Thesis, 2001.
%F 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
%p 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):
%p S:= series(gf,x,103):
%p seq(coeff(S,x,j),j=0..100); # _Robert Israel_, Dec 08 2014
%Y Cf. A002898, A057647.
%K nonn
%O 0,3
%A _Cyril Banderier_, Oct 12 2000
%E Title corrected by _Sean A. Irvine_, Jun 22 2022
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