On the number of hexagonal polyominoes

AbstractA 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), pn with n hexagons. We obtain pn for n⩽35. We prove that pn=τn+o(n), obtain the bounds 4.8049⩽τ⩽5.9047, and estimate that τ=5.1831478(17). Finally, we provide compelling numerical evidence that the generating function ∑pnzn≈A(z)log(1−τz), for z→(1/τ)− with A(z) holomorphic in a cut plane, estimate A(1/τ) and predict the sub-leading asymptotic behaviour, identifying a non-analytic correction-to-scaling term with exponent Δ=3/2. On the basis of universality and previous numerical work we argue that the mean-square radius of gyration 〈Rg2〉n of polyominoes of size n grows as n2ν, with ν=0.64115(5)