The maximal order of Stern's diatomic sequence

Abstract

We answer a question of Calkin and Wilf concerning the maximal order of Stern's diatomic sequence. Specifically, we prove that $$\limsup_{n\to\infty}\frac{a(n)}{\varphi^{\log_2 n}}=\frac{\varphi^{\log_2 3}}{\sqrt{5}},$$ where $\varphi=(\sqrt{5}+1)/2$ is the golden ratio. This improves on previous results given by Berlekamp, Conway, and Guy, who showed that the limit value was bounded above by 1.25, and by Calkin and Wilf, who showed that the exact value was in the interval $[({\varphi}/{\sqrt{5}})({3}/{2})^{\log_2\varphi},(\varphi+1)/{\sqrt{5}}].$