A379178 - OEIS

%I #10 Dec 23 2024 22:19:42

%S 6,18,90,479,2718,16126,97885,603741,3771287,23792622,151342506,

%T 969465873,6248109573

%N Number of fixed site animals with n nodes on the nodes of the kisrhombille tiling.

%C Site animals on a lattice (regular graph) are connected induced subgraphs up to translation.

%C Dual to the site animals on the nodes of the truncated trihexagonal tiling, counted by A197464, insofar as the tilings are each others' duals.

%C The Madras reference gives a good treatment of site animals on general lattices.

%C It is a consequence of the Madras work that lim_{n\to\infty} a(n+1)/a(n) converges to some growth constant c.

%C Terms a(1)-a(13) 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.

%D Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

%H Anthony J. Guttman (Ed.), <a href="https://doi.org/10.1007/978-1-4020-9927-4">Polygons, Polyominoes, and Polycubes</a>, Canopus Academic Publishing Limited, Bristol, 2009.

%H Iwan Jensen, <a href="https://doi.org/10.1023/A:1004855020556">Enumerations of Lattice Animals and Trees</a>, Journal of Statistical Physics 102 (2001), 865-881.

%H N. Madras, <a href="https://doi.org/10.1007/BF01608793">A pattern theorem for lattice clusters</a>, Annals of Combinatorics, 3 (1999), 357-384.

%H N. Madras and G. Slade, <a href="https://doi.org/10.1007/978-1-4614-6025-1">The Self-Avoiding Walk</a>, Birkhäuser Publishing (1996).

%H D. Hugh Redelmeier, <a href="https://doi.org/10.1016/0012-365X(81)90237-5">Counting Polyominoes: Yet Another Attack</a>, Discrete Mathematics 36 (1981), 191-203.

%H Markus Vöge and Anthony J. Guttman, <a href="https://doi.org/10.1016/S0304-3975(03)00229-9">On the number of hexagonal polyominoes</a>. Theoretical Computer Science, 307 (2003), 433-453.

%F 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.

%e There are 6 translationally distinct sites in the kisrhombille lattice, so a(1)=6.

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

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

%K nonn,hard,more

%O 1,1

%A _Johann Peters_, Dec 17 2024