0,3

Also, number of closed paths of length n on the honeycomb lattice.

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

Contribution from David Callan (callan(AT)stat.wisc.edu), Aug 25 2009: (Start)

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

12..13..21..23..31..32

21..31..12..32..13..23. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.

Cf. solution to 1995 Putnam problem A-6, Am. Math. Monthly, 1996, p. 674.

C. Banderier, Analytic combinatorics of random walks and planar maps, PhD Thesis, 2001.

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

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.

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.

Sequence in context: A054883 A005402 A128953 this_sequence A003613 A099767 A080450

Adjacent sequences: A002895 A002896 A002897 this_sequence A002899 A002900 A002901

nonn,walk,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from David Bloom 3/97.

Formula and further terms from Cyril Banderier (Cyril.Banderier(AT)inria.fr), Oct 12 2000