On a sequence of Kimberling and its relationship to the Tribonacci word
🤖 AI Summary
This paper resolves several central conjectures posed by Clark Kimberling (2017) concerning the binary sequence $mathbf{B} = 0100101100cdots$, generated by a specific morphic substitution system and closely related to the infinite Tribonacci word. Adopting a formal automata-theoretic approach, we employ the Walnut theorem-proving tool to rigorously verify combinatorial properties of $mathbf{B}$ for the first time. Our results establish that the factor complexity of $mathbf{B}$ is exactly $2n+1$ for all $n geq 1$, its critical exponent equals $2$, and there exists a computable bijection between $mathbf{B}$ and the infinite Tribonacci word. These contributions not only confirm Kimberling’s conjectures but also advance the verification of structural properties of infinite words to a mechanized, formally verifiable level. The work provides a new paradigm for combinatorial analysis of automatic sequences through automated reasoning.
Application Category
📝 Abstract
In 2017, Clark Kimberling defined an interesting sequence ${f B} = 0100101100 cdots$ 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 $f B$.
Problem
Research questions and friction points this paper is trying to address.
Proving conjectures about Kimberling's binary sequence
Relating the sequence to the infinite Tribonacci word
Determining subword complexity and critical exponent
Innovation
Methods, ideas, or system contributions that make the work stand out.
Used Walnut theorem-prover for verification
Linked Kimberling sequence to Tribonacci word
Analyzed subword complexity and critical exponent
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