Palindromic sequences of the Markov spectrum

TL;DR

This paper investigates the palindromic nature of Markov sequences derived from continued fractions in the Markov spectrum, revealing a connection with Stern's diatomic sequence and circular shifts.

Contribution

It establishes that the periods of Markov sequences become palindromic after a specific number of circular shifts determined by Stern's diatomic sequence.

Findings

Periods are palindromic after shifts

Number of shifts linked to Stern's sequence

Provides insight into structure of Markov spectrum sequences

Abstract

We study the periods of Markov sequences, which are derived from the continued fraction expression of elements in the Markov spectrum. This spectrum is the set of minimal values of indefinite binary quadratic forms that are specially normalised. We show that the periods of these sequences are palindromic after a number of circular shifts, the number of shifts being given by Stern's diatomic sequence.