An improved upper bound on the growth constant of polyiamonds

Abstract

A polyiamond is an edge-connected set of cells on the triangular lattice.  Let~$T(n)$ denote the number of distinct (up to translation) polyiamonds made of~$n$ cells. It is known that the sequence~$T(n)$ has an asymptotic growth constant, i.e., the limit $\lambda_T := \lim_{n \to \infty} T(n+1) / T(n)$ exists, but the exact value of~$\lambda_T$ is still unknown.  In this paper, we improve considerably the best known upper bound on~$\lambda_T$ from~4 to~3.6108.