Record-setters in the Stern sequence

Abstract

Stern's diatomic series, denoted by (a(n))n≥0, is defined by the recurrence relations a(2n)=a(n) and a(2n+1)=a(n)+a(n+1) for n≥1, with initial values a(0)=0 and a(1)=1. A record-setter for a sequence (s(n))n≥0 is an index v such that s(i)<s(v) holds for all i<v. In this paper, we provide a complete description of the record-setters for the Stern sequence. As a consequence, we prove that the running max sequence of the Stern sequence is not 2-regular.