A378416 Number of fixed site animals with n nodes on the nodes of the rhombille tiling.

3, 6, 21, 73, 273, 1049, 4117, 16416, 66263, 270211, 1111443, 4605575, 19204920, 80515734, 339137432, 1434319849

Offset: 1

Author

Johann Peters, Nov 25 2024

Keywords

• nonn
• hard
• more

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 Guttman 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. Guttman (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.
• N. Madras, A pattern theorem for lattice clustersA pattern theorem for lattice clusters, Annals of Combinatorics, 3 (1999), 357-384.
• N. Madras and G. 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. Guttman, On the number of hexagonal polyominoes. Theoretical Computer Science, 307 (2003), 433-453.

Crossrefs

The platonic tilings are associated with the following sequences: square A001168; triangular A001207; and hexagonal A001420.

The other 8 isogonal tilings are associated with these, A197160, A197158, A196991, A196992, A197461, A196993, A197464, A197467.

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.