A003202 Cluster series for hexagonal lattice.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• more

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

• 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).

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