%I A002898 M4101 N1701
%S A002898 1,0,6,12,90,360,2040,10080,54810,290640,1588356,8676360,47977776,
%T A002898 266378112,1488801600,8355739392,47104393050,266482019232,
%U A002898 1512589408044,8610448069080,49144928795820,281164160225520
%N A002898 Number of n-step closed paths on hexagonal lattice.
%C A002898 Also, number of closed paths of length n on the honeycomb lattice.
%C A002898 The hexagonal lattice is the familiar 2-dimensional lattice in which 
               each point has 6 neighbors. This is sometimes called the triangular 
               lattice.
%C A002898 Contribution from David Callan (callan(AT)stat.wisc.edu), Aug 25 2009: 
               (Start)
%C A002898 a(n) = number of 2-by-n matrices, entries from {1,2,3}, second row a 
               (multiset) permutation of the first, with no constant columns. For 
               example, a(2)=6 counts the matrices
%C A002898 12..13..21..23..31..32
%C A002898 21..31..12..32..13..23. (End)
%D A002898 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002898 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002898 C. Domb, On the theory of cooperative phenomena in crystals, Advances 
               in Phys., 9 (1960), 149-361.
%D A002898 Cf. solution to 1995 Putnam problem A-6, Am. Math. Monthly, 1996, p. 
               674.
%H A002898 C. Banderier, <a href="http://algo.inria.fr/banderier/">Analytic combinatorics 
               of random walks and planar maps</a>, PhD Thesis, 2001.
%H A002898 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               lattices/A2.html">Home page for hexagonal (or triangular) lattice 
               A2</a>
%F A002898 a(0) = 1, a(1) = 0, a(2)=6, (108*n+72+36*n^2)*a(n)+(24*n^2+96*n+96)*a(n+1)+(n^2+5*n+6)*a(n+2)+(-6*n-9-n^2)*a(\
               n+3)=0.
%F A002898 E.g.f.: (BesselI(0,2*x))^3+2*sum((BesselI(k,2*x))^3,k=1..infinity), from 
               Karol A. Penson (penson(AT)lptl.jussieu.fr) Aug 18 2006.
%Y A002898 Sequence in context: A054883 A005402 A128953 this_sequence A003613 A099767 
               A080450
%Y A002898 Adjacent sequences: A002895 A002896 A002897 this_sequence A002899 A002900 
               A002901
%K A002898 nonn,walk,nice
%O A002898 0,3
%A A002898 N. J. A. Sloane (njas(AT)research.att.com).
%E A002898 More terms from David Bloom 3/97.
%E A002898 Formula and further terms from Cyril Banderier (Cyril.Banderier(AT)inria.fr), 
               Oct 12 2000