OFFSET
1,1
COMMENTS
Site animals on a lattice (regular graph) are connected induced subgraphs up to translation.
Dual to the site animals on the nodes of the trihexagonal (AKA kagome) tiling, counted by A197461, insofar as the tilings are each others' duals.
The Madras reference gives a good treatment of site animals on general lattices.
It is a consequence of the Madras work that lim_{n\to\infty} a(n+1)/a(n) converges to some growth constant c.
Terms a(1)-a(16) were found by running a generalization of Redelmeier's algorithm. The transfer matrix algorithm (TMA) is more efficient than Redelmeier's for calculating regular polyominoes, and may give more terms here too. See the Jensen reference for a treatment of the TMA. See the Vöge and Guttmann reference for an implementation of the TMA on the triangular lattice to count polyhexes, A001207.
REFERENCES
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.
LINKS
Anthony J. Guttmann (Ed.), Polygons, Polyominoes and Polycubes, Canopus Academic Publishing Limited, Bristol, 2009.
Iwan Jensen, Enumerations of Lattice Animals and Trees, Journal of Statistical Physics 102 (2001), 865-881.
Neal Madras, A pattern theorem for lattice clusters, Annals of Combinatorics, 3 (1999), 357-384.
Neal Madras and Gordon Slade, The Self-Avoiding Walk, Birkhäuser Publishing (1996).
D. Hugh Redelmeier, Counting Polyominoes: Yet Another Attack, Discrete Mathematics 36 (1981), 191-203.
Markus Vöge and Anthony J. Guttmann, On the number of hexagonal polyominoes, Theoretical Computer Science, 307 (2003), 433-453.
FORMULA
It is widely believed site animals on 2-dimensional lattices grow asymptotically to kc^n/n, where k is a constant and c is the growth constant, dependent only on the lattice. See the Madras and Slade reference.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Johann Peters, Nov 25 2024
STATUS
approved