cp's OEIS Frontend

A378344 Number of fixed site animals with n nodes on the nodes of the prismatic pentagonal tiling.

%I A378344 #19 Jan 03 2025 23:28:30
%S A378344 3,5,12,35,106,332,1062,3466,11496,38621,131042,448146,1542548,
%T A378344 5338641,18563680,64814950,227117365,798387748,2814618634
%N A378344 Number of fixed site animals with n nodes on the nodes of the prismatic pentagonal tiling.
%C A378344 Site animals on a lattice (regular graph) are connected induced subgraphs up to translation.
%C A378344 Dual to the polyhouses, AKA the site animals on the nodes of the elongated triangular tiling, counted by A197158, insofar as the tilings are each others' duals.
%C A378344 The Madras reference gives a good treatment of site animals on general lattices.
%C A378344 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 A378344 Terms a(1)-a(19) 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 A378344 Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.
%H A378344 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 A378344 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 A378344 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 A378344 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 A378344 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 A378344 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 A378344 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.
%Y A378344 The platonic tilings are associated with the following sequences: square A001168; triangular A001207; and hexagonal A001420.
%Y A378344 The other 8 isogonal tilings are associated with these, A197160, A197158, A196991, A196992, A197461, A196993, A197464, A197467.
%K A378344 nonn,hard,more
%O A378344 1,1
%A A378344 _Johann Peters_, Nov 23 2024