## Michel Rigo ; Manon Stipulanti - Automatic sequences: from rational bases to trees

dmtcs:8455 - Discrete Mathematics & Theoretical Computer Science, July 19, 2022, vol. 24, no. 1 - https://doi.org/10.46298/dmtcs.8455
Automatic sequences: from rational bases to trees

Authors: Michel Rigo ; Manon Stipulanti

The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration system with a regular numeration language, we consider those built on languages associated with trees having periodic labeled signatures and, in particular, rational base numeration systems. We obtain two main characterizations of these sequences. The first one is concerned with $r$-block substitutions where $r$ morphisms are applied periodically. In particular, we provide examples of such sequences that are not morphic. The second characterization involves the factors, or subtrees of finite height, of the tree associated with the numeration system and decorated by the terms of the sequence.

Volume: vol. 24, no. 1
Section: Automata, Logic and Semantics
Published on: July 19, 2022
Accepted on: May 31, 2022
Submitted on: September 7, 2021
Keywords: Computer Science - Formal Languages and Automata Theory,Computer Science - Discrete Mathematics,Mathematics - Combinatorics,68R15, 68Q45, 11A63