Record-Setters in the Stern Sequence

TL;DR

This paper provides a comprehensive characterization of the indices in the Stern sequence where new maximum values are recorded, enhancing understanding of its extremal behavior.

Contribution

It offers a complete description of record-setters in the Stern sequence, a detailed analysis not previously available.

Findings

Identifies all record-setters in the Stern sequence.

Provides a characterization of the positions where new maxima occur.

Enhances understanding of the sequence's extremal properties.

Abstract

Stern's diatomic series, denoted by $(a(n))_{n \geq 0}$, is defined by the recurrence relations $a(2n) = a(n)$ and $a(2n + 1) = a(n) + a(n + 1)$ for $n \geq 1$, and initial values $a(0) = 0$ and $a(1) = 1$. A record-setter for a sequence $(s(n))_{n \geq 0}$ is an index $v$ such that $s(i) < s(v)$ holds for all $i < v$. In this paper, we give a complete description of the record-setters for the Stern sequence.