On a sequence of Kimberling and its relationship to the Tribonacci word
arXiv:2510.1131828.4h-index: 6
Predicted impact top 38% in CO · last 90 daysOriginality Synthesis-oriented
AI Analysis
This resolves specific conjectures in combinatorics on words, but it is incremental as it builds on known sequences and methods.
The paper proves conjectures about the Kimberling sequence by relating it to the infinite Tribonacci word, determining its subword complexity and critical exponent.
In 2017, Clark Kimberling defined an interesting sequence ${\bf B} = 0100101100 \cdots$ 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 $\bf B$.