Predicting the Number of Hexagonalsystems with 24 and 25 Hexagons

We predict the number of hexagonal systems consisting of 24 and 25 hexagons to be H 24 = 122237774262384 and H 25 = 606259305418149, with 6 and 5 signiicant digits, respectively. Further estimates for H n up to n = 31 are also given. Informally speaking, a hexagonal system can be viewed as a connected arrangement of hexagonal cells packed in the same way as the typical honeycomb arrangement in a beehive. More formally, it is a nite connected plane graph with no cut-vertices, in which all interior regions are mutually congruent regular hexagons 1]. Hexagonal systems have from time to time attracted the attention of mathematicians (and were named \hexagonal animals", \honeycomb systems", \polyhexes", etc.), in connection with statistical physics and applications to lattice gas models 2, 3, 4]. But the main interest in them comes from chemistry: hexagonal systems are the natural graph representations of benzenoid hydrocarbons , whence the names \benzenoid graphs", \benzenoid systems", and \fusenes" used in the chemical literature. An enormous literature exists on various chemical applications of hexagonal systems. We refer to 5, 6] for details and references. One of the classical problems in the theory of hexagonal systems is their enumeration. In what follows, the number of non-isomorphic hexagonal systems consisting of n hexagons is denoted by H n , where \non-isomorphic" means viewed up to translations, rotations, and symmetries. This in turn is equal to the number of n-cyclic benzenoid hydrocarbons. The rst few values of H n are given in Table 1. The enumeration of hexagonal systems according to area stands as one of the most challenging unsolved problems of combinatorial theory (cf. Section 10.8.5 in 7]). In spite of numerous attempts, no one was successful in applying PP olya's theory 7, 8, 9] or any other technique of combinatorics to nd H n or, at least, in establishing the asymptotic behavior n 1 2 3 4 5 6 H n