Largest values of the Stern sequence, alternating binary expansions and continuants

TL;DR

This paper investigates the maximum values in Stern's diatomic array by connecting it with alternating binary expansions and continuants, leading to new ordering results and proofs of existing conjectures.

Contribution

It introduces a novel approach linking Stern sequence, binary expansions, and continuants, and solves conjectures about the largest values in the sequence.

Findings

Proved conjectures of Lansing regarding the largest values.

Established a method to order continuants of special shape.

Connected Stern sequence analysis with Fibonacci number identities.

Abstract

We study the largest values of the $r$th row of Stern's diatomic array. In particular, we prove some conjectures of Lansing. Our main tool is the connection between the Stern sequence, alternating binary expansions and continuants. This allows us to reduce the problem of ordering the elements of the Stern sequence to the problem of ordering continuants. We describe an operation that increases the value of a continuant, allowing us to reduce the problem of largest continuants to ordering continuants of very special shape. Finally, we order these special continuants using some identities and inequalities involving Fibonacci numbers.