Proof of Northshield's conjecture concerning an analogue of Stern's sequence for $\mathbb{Z}[\sqrt{2}]$

Abstract

We prove a conjecture of Northshield by determining the maximal order of his analogue of Stern's sequence for $\mathbb{Z}[\sqrt{2}]$. In particular, if $b$ is Northshield's analogue, we prove that $$\underset{n\to \infty }{\mathrm{limsup}}\frac{2b(n)}{(2n{)}^{{\log }_3(\sqrt{2}+1)}}=1.$$

6 pages