Automatic sequences: from rational bases to trees

Abstract

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.

The Thue-Morse sequence in base 3/2

“…Next we consider the question whether also the sequence y 3/2 , the sum of digits function modulo 2 of the base 1/2•3/2 representation, is fixed point of a 2-block substitution. This is indeed the case, and this 2-block substitution is given by Rigo and Stipulanti in [8]. In [8] the proof of Theorem 10 is based on a generalization of Cobham's theorem to what are called S-automatic sequences built on tree languages with a periodic labeled signature.…”

“…This is indeed the case, and this 2-block substitution is given by Rigo and Stipulanti in [8]. In [8] the proof of Theorem 10 is based on a generalization of Cobham's theorem to what are called S-automatic sequences built on tree languages with a periodic labeled signature. Here we consider a more direct route, based on a simple closure property of p-q-block-substitutions.…”

“…Many authors refer to the paper [1] from Akiyama, Frougny, and Sakarovitch for the properties of base 3/2 expansions (see, e.g., [7], [8]). However, the p/q expansions studied in paper [1] are different from the 3/2 expansions that are usually considered as in Equation ( 1).…”

“…be the sum of digits function of the base 3/2 expansions. We have (see A244040 in [6]) 3,4,3,4,5,3,4,5,5,6,7,4,5,6,5,6,7,7,8,9,5,6,7,5,6,7,7,8, 9, 8, 9, 10, . .…”