Bounding Klarner's constant from above using a simple recurrence

TL;DR

This paper presents a simplified recurrence-based method to upper bound Klarner's constant, improving understanding of polyomino growth rates with a new, straightforward proof.

Contribution

The paper introduces a simpler recurrence relation to bound the number of polyominoes, providing an alternative proof to existing bounds on Klarner's constant.

Findings

Established an upper bound of approximately 4.83 for Klarner's constant.

Derived a recurrence relation for the maximum number of polyominoes with n cells.

Provided a new combinatorial interpretation of the sequence G(n).

Abstract

Klarner and Rivest showed that the growth of the number of polyominoes, also known as Klarner's constant, is at most $2+2\sqrt{2}<4.83$ by viewing polyominoes as a sequence of twigs with appropriate weights given to each twig and studying the corresponding multivariate generating function. In this short note, we give a simpler proof by a recurrence on an upper bound. In particular, we show that the number of polyominoes with $n$ cells is at most $G(n)$ with $G(0)=G(1)=1$ and for $n\ge 2$, \[ G(n) = 2\sum_{m=1}^{n-1} G(m)G(n-1-m). \] It should be noted that $G(n)$ has multiple combinatorial interpretations in literature.