A003202 Cluster series for hexagonal lattice. (Formerly M4117)

1, 6, 18, 48, 126, 300, 750, 1686, 4074, 8868, 20892, 44634, 103392, 216348, 499908, 1017780, 2383596, 4648470, 11271102, 20763036, 52671018, 91377918

Offset

0,2

Comments

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

References

J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol. 2; see especially pp. 225-226.

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

Links

Table of n, a(n) for n=0..21.

S. Mertens, Lattice animals: a fast enumeration algorithm and new perimeter polynomials, J. Stat. Phys. 58 (1990) 1095-1108.

Stephan Mertens and Markus E. Lautenbacher, Counting lattice animals: A parallel attack, J. Stat. Phys., 66 (1992), 669-678.

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

M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315.

M. F. Sykes and Sylvia Flesia, Lattice animals: Supplementation of perimeter polynomial data by graph-theoretic methods, Journal of Statistical Physics, 63 (1991), 487-489.

M. F. Sykes and M. Glen, Percolation processes in two dimensions. I. Low-density series expansions, J. Phys. A: Math. Gen., 9 (1976), 87-95.

Crossrefs

Cf. A003203 (square net), A003204 (honeycomb net), A003197 (bond percolation).

Sequence in context: A248462 A256010 A128543 * A003198 A099857 A163765

Adjacent sequences: A003199 A003200 A003201 * A003203 A003204 A003205

Keyword

nonn,more

Author

N. J. A. Sloane

Extensions

a(10)-a(11) from Sean A. Irvine, Aug 16 2020

a(12)-a(18) added from Mertens by Andrey Zabolotskiy, Feb 01 2022

a(19)-a(21) from Mertens & Lautenbacher added by Andrey Zabolotskiy, Jan 28 2023

Status

approved