Beyond odious and evil

Allouche, Jean‐Paul; Cloitre, Benoit; Shevelev, Vladimir

doi:10.1007/s00010-015-0345-3

Cited by 12 publications

(14 citation statements)

References 9 publications

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“…n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 While writing this paper, we realized that a result similar to Theorem 6 and to Theorem 7 was proved in [1], where the paperfolding sequence was replaced with a generalized Thue-Morse sequence. For example, let (t n ) n≥0 be the usual Thue-Morse sequence (see, e.g., [3]), where t n is the parity of the sum of the binary digits of n. Let (u n ) n≥0 denote the increasing sequence of odious numbers, namely the numbers n for which t n = 1, and (v n ) n≥0 be the increasing sequence of evil numbers, namely the numbers n for which t n = 0.…”

Section: On the First Three Conjectures Of Barrymentioning

confidence: 88%

See 1 more Smart Citation

On some conjectures of P. Barry

Allouche,

Han,

Shallit

2020

Preprint

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Section: On the First Three Conjectures Of Barrymentioning

confidence: 88%

“…If the sequence (a n ) n≥0 were q-regular, its difference sequence, say (d n ) n≥0 , where d n = a n+1 − a n , would be q-regular as well. But (d n ) n≥0 takes only finitely many values: recall that all integers congruent to 1 modulo 4 are values of a n , so that d n ∈ [1,4]. Thus, if (d n ) n≥0 were q-regular, it would be q-automatic.…”

mentioning

confidence: 99%

On some conjectures of P. Barry

Allouche,

Han,

Shallit

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…where d and d are given by (3). Moreover, note such d and d are always greater than or equal to − r m , hence they are greater than −1.…”

Section: States Of the Same Class Are Indistinguishablementioning

confidence: 99%

“…This infinite word is one of the archetypical aperiodic automatic words, see the surveys [6,34]. Many number-theoretic works devoted to sets of integers defined thanks to the Thue-Morse word exist, such as the study of additive and multiplicative properties, or iterations and sums of such sets [3,12,30]. In this vein, the set T seems to be a natural candidate to start with.…”

Section: Introductionmentioning

confidence: 99%

Minimal Automaton for Multiplying and Translating the Thue-Morse Set

Charlier

Cisternino

Massuir

2021

Electron. J. Combin.

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The Thue-Morse set $\mathcal{T}$ is the set of those non-negative integers whose binary expansions have an even number of $1$'s. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word $${\tt 0110100110010110\cdots},$$ which is the fixed point starting with ${\tt 0}$ of the word morphism ${\tt 0\mapsto 01}$, ${\tt 1\mapsto 10}$. The numbers in $\mathcal{T}$ are commonly called the evil numbers. We obtain an exact formula for the state complexity of the set $m\mathcal{T}+r$ (i.e. the number of states of its minimal automaton) with respect to any base $b$ which is a power of $2$. Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all $2^p$-expansions of the set of integers $m\mathcal{T}+r$ for any positive integers $p$ and $m$ and any remainder $r\in\{0,\ldots,m{-}1\}$. The proposed method is general for any $b$-recognizable set of integers.

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“…This infinite word is one of the archetypical aperiodic automatic words, see the surveys [6,35]. Many number-theoretic works devoted to sets of integers defined thanks to the Thue-Morse word exist, such as the study of additive and multiplicative properties, or iterations and sums of such sets [3,12,31]. In this vein, the set T seems to be a natural candidate to start with.…”

Section: Introductionmentioning

confidence: 99%

Minimal automaton for multiplying and translating the Thue-Morse set

Charlier¹,

Cisternino²,

Massuir³

2019

Preprint

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The Thue-Morse set T is the set of those non-negative integers whose binary expansions have an even number of 1. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word abbabaabbaababba • • •, which is the fixed point starting with a of the word morphism a → ab, b → ba. The numbers in T are commonly called the evil numbers. We obtain an exact formula for the state complexity of the set mT + r (i.e. the number of states of its minimal automaton) with respect to any base b which is a power of 2. Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all 2 p -expansions of the set of integers mT + r for any positive integers p and m and any remainder r ∈ {0, . . . , m − 1}. The proposed method is general for any b-recognizable set of integers. As an application, we obtain a decision procedure running in quadratic time for the problem of deciding whether a given 2 p -recognizable set is equal to a set of the form mT + r.

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Beyond odious and evil

Cited by 12 publications

References 9 publications

On some conjectures of P. Barry

On some conjectures of P. Barry

Minimal Automaton for Multiplying and Translating the Thue-Morse Set

Minimal automaton for multiplying and translating the Thue-Morse set

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