On a sequence of Kimberling and its relationship to the Tribonacci word

Abstract

In 2017, Clark Kimberling defined an interesting sequence $\mathbf{B}=0100101100\cdots$ of $0$'s and $1$'s by certain inflation rules, and he made a number of conjectures about this sequence and some related ones. In this note we prove his conjectures using, in part, the Walnut theorem-prover. We show how his word is related to the infinite Tribonacci word, and we determine both the subword complexity and critical exponent of $\mathbf{B}$.