Bounding Klarner’s constant from above using a simple recurrence

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 \(n\ge 2\),

$$\begin{aligned} G(n) = 2\sum _{m=1}^{n-1} G(m)G(n-1-m). \end{aligned}$$

It should be noted that G(n) has multiple combinatorial interpretations in the literature.