On the number of hexagonal polyominoes
A combination of the refined finite lattice method and transfer matrices allows a radical increase in the computer enumeration of polyominoes on the hexagonal lattice (equivalently, site clusters on the triangular lattice), p(n) with n hexagons. We obtain p(n) for n less than or equal to 35. We prove that p(n) = tau(n+o(n)), obtain the bounds 4.8049 less than or equal to tau less than or equal to 5.9047, and estimate that tau = 5.1831478(17). Finally, we provide compelling numerical evidence that the generating function Sigma p(n)z(n) approximate to A(z)log(1 - tauz), for z --> (1/tau)(-) with A(z) holomorphic in a cut plane, estimate A(1/tau) and predict the sub-leading asymptotic behaviour, identifying a non-analytic correction-to-scaling term with exponent Delta = 3/2. On the basis of universality and previous numerical work we argue that the mean-square radius of gyration (R-g(2))(n) of polyominoes of size n grows as n(2v), with v = 0.64115(5). (C) 2003 Elsevier B.V. All rights reserved.
polyominoes enumeration hexagonal lattice site clusters triangular lattice