An Improved Upper Bound on the Growth Constant of Polyominoes

Polyominoes are edge-connected sets of squares on the square lattice. The symbol λ usually denotes the growth constant of A(n), the sequence that enumerates polyominoes. In this paper we prove that λ ≤ 4.5685 by analyzing the growth constant of a sequence B(n), for which B(n) ≥ A(n) for any value of n ∈ N. The recursive formula for B(n) is based on the representation of a polyomino as the assembly of a pair of smaller polyominoes and a code that describes the assembly. Then, an upper bound on the growth constant of B(n) is derived by a careful analysis of this assembly.