This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A057648 #22 Apr 30 2024 03:50:17 %S A057648 1,0,2,2,13,34,158,594,2665,11558,53320,247488,1181266,5708884, %T A057648 28049474,139417402,701063005,3559326294,18233244530,94140532624, %U A057648 489573775236,2562613997512,13493827469116,71441865994904 %N A057648 Number of excursions of length n on the upper-right part of the hexagonal lattice. %C A057648 Excursions = walks from the origin to the origin. %C A057648 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 A057648 Robert Israel, <a href="/A057648/b057648.txt">Table of n, a(n) for n = 0..1000</a> %H A057648 C. Banderier, <a href="http://algo.inria.fr/banderier/">Analytic combinatorics of random walks and planar maps</a>, PhD Thesis, 2001. %F A057648 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 %F A057648 a(n) ~ (2*sqrt(3) - 3) * 2^n * 3^(n+2) / (Pi*n^3). - _Vaclav Kotesovec_, Apr 30 2024 %p A057648 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 A057648 S:= series(gf,x,103): %p A057648 seq(coeff(S,x,j),j=0..100); # _Robert Israel_, Dec 08 2014 %Y A057648 Cf. A002898, A057647. %K A057648 nonn %O A057648 0,3 %A A057648 _Cyril Banderier_, Oct 12 2000 %E A057648 Title corrected by _Sean A. Irvine_, Jun 22 2022