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

TL;DR

This paper proves Northshield's conjecture by determining the maximal growth order of an analogue of Stern's sequence over the ring 2, establishing a precise limit superior for its scaled values.

Contribution

It provides a rigorous proof of Northshield's conjecture, identifying the exact asymptotic maximal order of his sequence analogue over 2.

Findings

Established the exact limit superior of the sequence's scaled values.

Confirmed the conjectured growth rate of Northshield's sequence analogue.

Contributed to understanding the behavior of sequences over algebraic integers.

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 $$\limsup_{n\to\infty}\frac{2b(n)}{(2n)^{\log_3 (\sqrt{2}+1)}}=1.$$