On the Number of Unbordered Factors

TL;DR

This paper introduces a technique for counting unbordered factors in automatic sequences, proves a conjecture about the Thue-Morse sequence, and explores properties of unbordered factors in various sequences.

Contribution

It presents a general enumeration method for factors of automatic sequences and proves a key conjecture about the number of unbordered factors in the Thue-Morse sequence.

Findings

f(n) <= n for n >= 4 in the Thue-Morse sequence

f(n) = n infinitely often in the Thue-Morse sequence

automatic sequences can have exactly 2 unbordered factors of every length

Abstract

We illustrate a general technique for enumerating factors of k-automatic sequences by proving a conjecture on the number f(n) of unbordered factors of the Thue-Morse sequence. We show that f(n) <= n for n >= 4 and that f(n) = n infinitely often. We also give examples of automatic sequences having exactly 2 unbordered factors of every length.