cp's OEIS Frontend

A378362 Number of fixed site animals containing n nodes on the nodes of the cairo pentagonal tiling.

%I A378362 #19 Dec 17 2024 13:04:48
%S A378362 6,10,24,68,198,594,1816,5650,17824,56836,182788,592060,1929676,
%T A378362 6323418,20819284,68828316,228372578,760188362,2537770576,8494004948
%N A378362 Number of fixed site animals containing n nodes on the nodes of the cairo pentagonal tiling.
%C A378362 Site animals on a lattice (regular graph) are connected induced subgraphs up to translation.
%C A378362 Is dual to the polycairos counted by A196991, AKA site animals on the nodes of the snub square tiling, insofar that these tilings are each others' duals.
%C A378362 The Madras reference gives a good treatment of site animals on general lattices.
%C A378362 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 A378362 Terms a(1)-a(20) 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 A378362 Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.
%H A378362 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 A378362 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 A378362 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 A378362 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 A378362 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 A378362 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 A378362 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 A378362 There are six translationally distinct nodes in the cairo pentagonal tiling, so a(1)=6.
%Y A378362 The platonic tilings are associated with the following sequences: square A001168; triangular A001207; and hexagonal A001420.
%Y A378362 The other 8 isogonal tilings are associated with these, A197160, A197158, A196991, A196992, A197461, A196993, A197464, A197467.
%K A378362 nonn,hard,more
%O A378362 1,1
%A A378362 _Johann Peters_, Nov 23 2024