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See detailThe Formal Inverse of the Period-Doubling Sequence
Rampersad, Narad; Stipulanti, Manon ULiege

in Journal of Integer Sequences (2018), 21(9), 189122

If $p$ is a prime number, consider a p-automatic sequence $(u_n)_{n\ge 0}$, and let $U(X) = $\sum_{n\ge 0} u_nX^n ∈ F_p[[X]]$ be its generating function. Assume that there exists a formal power series $V ... [more ▼]

If $p$ is a prime number, consider a p-automatic sequence $(u_n)_{n\ge 0}$, and let $U(X) = $\sum_{n\ge 0} u_nX^n ∈ F_p[[X]]$ be its generating function. Assume that there exists a formal power series $V(X) = \sum_{n\ge 0} v_n X^n ∈ F_p[[X]]$ which is the compositional inverse of $U$, i.e., $U(V(X)) = X = V(U(X))$. The problem investigated in this paper is to study the properties of the sequence $(v_n)_{n\ge 0}$. The work was first initiated for the Thue–Morse sequence, and more recently the case of other sequences (variations of the Baum-Sweet sequence, variations of the Rudin-Shapiro sequence and generalized Thue-Morse sequences) has been treated. In this paper, we deal with the case of the period-doubling sequence. We first show that the sequence of indices at which the period-doubling sequence takes the value 0 (resp., 1) is not k-regular for any $k \ge 2$. Secondly, we give recurrence relations for its formal inverse, then we show that it is 2-automatic, and we also provide an automaton that generates it. Thirdly, we study the sequence of indices at which this formal inverse takes the value 1, and we show that it is not k-regular for any $k \ge 2$ by connecting it to the characteristic sequence of Fibonacci numbers. We leave as an open problem the case of the sequence of indices at which this formal inverse takes the value 0. [less ▲]

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See detailThe constant of recognizability is computable for primitive morphisms
Durand, Fabien; Leroy, Julien ULiege

in Journal of Integer Sequences (2017), 20(4),

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See detailDo the properties of an $S$-adic representation determine factor complexity?
Durand, Fabien; Leroy, Julien ULiege; Richomme, Gwenaël

in Journal of Integer Sequences (2013), 16(2), 132630

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See detailSome properties of abelian return words
Rigo, Michel ULiege; Salimov, Pavel ULiege; Vandomme, Elise ULiege

in Journal of Integer Sequences (2013), 16

We investigate some properties of abelian return words as recently introduced by S. Puzynina and L. Q. Zamboni. In particular, we obtain a characterization of Sturmian words with non-null intercept in ... [more ▼]

We investigate some properties of abelian return words as recently introduced by S. Puzynina and L. Q. Zamboni. In particular, we obtain a characterization of Sturmian words with non-null intercept in terms of the finiteness of the set of abelian return words to all prefixes. We describe this set of abelian returns for the Fibonacci word but also for the 2-automatic Thue--Morse word. We also investigate the relationship existing between abelian complexity and finiteness of the set of abelian returns to all prefixes. We end this paper by considering the notion of abelian derived sequence. It turns out that, for the Thue--Morse word, the set of abelian derived sequences is infinite. [less ▲]

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See detailOn the Recognizability of Self-Generating Sets
Kärki, Tomi; Lacroix, Anne ULiege; Rigo, Michel ULiege

in Journal of Integer Sequences (2010), 13

Let I be a finite set of integers and F be a finite set of maps of the form n->k_i n + l_i with integer coefficients. For an integer base k>=2, we study the k-recognizability of the minimal set X of ... [more ▼]

Let I be a finite set of integers and F be a finite set of maps of the form n->k_i n + l_i with integer coefficients. For an integer base k>=2, we study the k-recognizability of the minimal set X of integers containing I and satisfying f(X)\subseteq X for all f in F. In particular, solving a conjecture of Allouche, Shallit and Skordev, we show under some technical conditions that if two of the constants k_i are multiplicatively independent, then X is not k-recognizable for any k>=2. [less ▲]

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