Predicting the Number of Hexagonal Systems with and Hexagons

We predict the number of hexagonal systems consisting of and hexagons to be H and H with and signi cant digits respectively Further estimates for Hn up to n are also given Hexagonal Systems 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 bee hive More formally it is a nite connected plane graph with no cut vertices in which all interior regions are mutually congruent regular hexagons 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 But the main interest in them comes from chemistry hexagonal systems are the natural graph representations of benzenoid hydro carbons 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 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 Hn 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 Hn are given in Table The enumeration of hexagonal systems according to area stands as one of the most challenging unsolved problems of combinatorial theory cf Section in In spite of numerous attempts no one was successful in applying P olya s theory or any other technique of combinatorics to nd Hn or at least in establishing the asymptotic behavior