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The ubiquitous ProuhetThueMorse sequence
 Sequences and their applications, Proceedings of SETA’98
, 1999
"... We discuss a wellknown binary sequence called the ThueMorse sequence, or the ProuhetThueMorse sequence. This sequence was introduced by Thue in 1906 and rediscovered by Morse in 1921. However, it was already implicit in an 1851 paper of Prouhet. The ProuhetThueMorse sequence appears to be som ..."
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Cited by 59 (9 self)
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We discuss a wellknown binary sequence called the ThueMorse sequence, or the ProuhetThueMorse sequence. This sequence was introduced by Thue in 1906 and rediscovered by Morse in 1921. However, it was already implicit in an 1851 paper of Prouhet. The ProuhetThueMorse sequence appears to be somewhat ubiquitous, and we describe many of its apparently unrelated occurrences.
Optimal Parallel Algorithms for Periods, Palindromes and Squares (Extended Abstract)
, 1992
"... ) Alberto Apostolico Purdue University and Universit`a di Padova Dany Breslauer yyz Columbia University Zvi Galil z Columbia University and TelAviv University Summary of results Optimal concurrentread concurrentwrite parallel algorithms for two problems are presented: ffl Finding all the pe ..."
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Cited by 32 (13 self)
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) Alberto Apostolico Purdue University and Universit`a di Padova Dany Breslauer yyz Columbia University Zvi Galil z Columbia University and TelAviv University Summary of results Optimal concurrentread concurrentwrite parallel algorithms for two problems are presented: ffl Finding all the periods of a string. The period of a string can be computed by previous efficient parallel algorithms only if it is shorter than half of the length of the string. Our new algorithm computes all the periods in optimal O(log log n) time, even if they are longer. The algorithm can be used to compute all initial palindromes of a string within the same bounds. ffl Testing if a string is squarefree. We present an optimal O(log log n) time algorithm for testing if a string is squarefree, improving the previous bound of O(log n) given by Apostolico [1] and Crochemore and Rytter [12]. We show matching lower bounds for the optimal parallel algorithms that solve the problems above on a general alphab...
Polynomial versus exponential growth in repetitionfree binary words
 To appear, J. Combinatorial Theory Ser. A
, 2003
"... It is known that the number of overlapfree binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7 3. More precisely, there are only polynomially many binary words of le ..."
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Cited by 18 (4 self)
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It is known that the number of overlapfree binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7 3. More precisely, there are only polynomially many binary words of length n that avoid 7 3powers, but there are exponentially many binary words of length n that avoid 7+ 3powers. This answers an open question of Kobayashi from 1986. 1
There Are Ternary Circular SquareFree Words of Length n for n ≥ 18
, 2002
"... There are circular squarefree words of length n on three symbols for n 18. This proves a conjecture of R. J. Simpson. ..."
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Cited by 14 (1 self)
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There are circular squarefree words of length n on three symbols for n 18. This proves a conjecture of R. J. Simpson.
The KomornikLoreti constant is transcendental
, 2000
"... ively 1) if the sum of the binary digits of n is even (respectively odd). This number q can be then obtained as the unique positive solution of 1 = P 1 n=1 ffi n q \Gamman . It is equal to 1:787231650::: In the electronic abstract of [4], the authors ask whether the number q = 1:787231650::: in ..."
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Cited by 13 (5 self)
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ively 1) if the sum of the binary digits of n is even (respectively odd). This number q can be then obtained as the unique positive solution of 1 = P 1 n=1 ffi n q \Gamman . It is equal to 1:787231650::: In the electronic abstract of [4], the authors ask whether the number q = 1:787231650::: in Theorem 1 above is irrational. The purpose of this note is to prove, as a simple consequence of a result of Mahler, that q is transcendental. Theorem 2 The number q = 1:787231650::: defined as the smallest number in (1; 2) for which there exists a unique expansion of 1 as 1 = P 1 n=1 ffi n q<F1
Counting OverlapFree Binary Words
 Springer LNCS 665
, 1993
"... A word on a finite alphabet A is said to be overlapfree if it contains no factor of the form xuxux, where x is a letter and u a (possibly empty) word. In this paper we study the number un of overlapfree binary words of length n, which is known to be bounded by a polynomial in n. First, we describe ..."
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Cited by 13 (1 self)
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A word on a finite alphabet A is said to be overlapfree if it contains no factor of the form xuxux, where x is a letter and u a (possibly empty) word. In this paper we study the number un of overlapfree binary words of length n, which is known to be bounded by a polynomial in n. First, we describe a bijection between the set of overlapfree words and a rational language. This yields recurrence relations for un , which allow to compute un in logarithmic time. Then, we prove that the numbers ff = sup f r j n r = O (un) g and fi = inf f r j un = O (n r ) g are distinct, and we give an upper bound for ff and a lower bound for fi. Finally, we compute an asymptotically tight bound to the number of overlapfree words of length less than n. 1 Introduction In general, the problem of evaluating the number un of words of length n in the language U consisting of words on some finite alphabet A with no factors in a certain set F is not easy. If F is finite, it amounts to counting words in...
The entropy of squarefree words
 Math. Comput. Modelling
, 1997
"... Finite alphabets of at least three letters permit the construction of squarefree words of infinite length. We show that the entropy density is strictly positive and derive reasonable lower and upper bounds. Finally, we present an approximate formula which is asymptotically exact with rapid converge ..."
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Cited by 12 (5 self)
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Finite alphabets of at least three letters permit the construction of squarefree words of infinite length. We show that the entropy density is strictly positive and derive reasonable lower and upper bounds. Finally, we present an approximate formula which is asymptotically exact with rapid convergence in the number of letters. Résumé Il est possible de construire des mots de longueur infinie sans carré sur un alphabet ayant au moins trois lettres. Nous démontrons que l’entropie du langage des mots sans carré sur un tel alphabet est strictement positive et l’encadrons par des bornes inférieure et supérieure raisonnables. Enfin, nous donnons pour l’entropie une expression approchée qui est asymptotiquement correcte et converge rapidement lorsque le nombre de lettres de l’alphabet tend vers l’infini.
An Optimal O(log log n) Time Parallel Algorithm for Detecting all Squares in a String
, 1995
"... An optimal O(log log n) time concurrentread concurrentwrite parallel algorithm for detecting all squares in a string is presented. A tight lower bound shows that over general alphabets this is the fastest possible optimal algorithm. When p processors are available the bounds become \Theta(d n ..."
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Cited by 11 (6 self)
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An optimal O(log log n) time concurrentread concurrentwrite parallel algorithm for detecting all squares in a string is presented. A tight lower bound shows that over general alphabets this is the fastest possible optimal algorithm. When p processors are available the bounds become \Theta(d n log n p e + log log d1+p=ne 2p). The algorithm uses an optimal parallel stringmatching algorithm together with periodicity properties to locate the squares within the input string.
Unavoidable Binary Patterns
 Acta Informatica
, 1993
"... Peter Roth proved that there are no binary patterns of length six or more that are unavoidable on the twoletter alphabet. He gave an almost complete description of unavoidable binary patterns. In this paper we prove one of his conjectures: the pattern ff 2 fi 2 ff is 2avoidable. From this we d ..."
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Cited by 10 (1 self)
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Peter Roth proved that there are no binary patterns of length six or more that are unavoidable on the twoletter alphabet. He gave an almost complete description of unavoidable binary patterns. In this paper we prove one of his conjectures: the pattern ff 2 fi 2 ff is 2avoidable. From this we deduce the complete classification of unavoidable binary patterns. We also study the concept of avoidability by iterated morphisms and prove that there are a few 2avoidable patterns which are not avoided by any iterated morphism. 1 Introduction The concept of unavoidable pattern was introduced by Bean, Ehrenfeucht & McNulty [2] and independently by Zimin [10]. They gave a characterization of unavoidable patterns, but there is still no characterization of kunavoidable patterns, i.e. patterns that are unavoidable over a kletter alphabet. However, we can try to find all kunavoidable patterns that can be written with a given alphabet. The case of unary patterns (in other terms: powers of a ...