no. 3
Squares and overlaps in the Thue-Morse sequence and some variants
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 3, pp. 473-484

We consider the position and number of occurrences of squares in the Thue-Morse sequence, and show that the corresponding sequences are 2-regular. We also prove that changing any finite but nonzero number of bits in the Thue-Morse sequence creates an overlap, and any linear subsequence of the Thue-Morse sequence (except those corresponding to decimation by a power of 2) contains an overlap.

DOI : 10.1051/ita:2006030
Classification : 68Q45, 68R15
Keywords: Thue-Morse word, overlap-free word, automatic sequence 
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     title = {Squares and overlaps in the {Thue-Morse} sequence and some variants},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
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Brown, Shandy; Rampersad, Narad; Shallit, Jeffrey; Vasiga, Troy. Squares and overlaps in the Thue-Morse sequence and some variants. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 3, pp. 473-484. doi: 10.1051/ita:2006030

[1] J.-P. Allouche, J. Currie, and J. Shallit, Extremal infinite overlap-free binary words. Electronic J. Combinatorics 5 (1998), #R27 (electronic). http://www.combinatorics.org/Volume_5/Abstracts/v5i1r27.html | Zbl | MR

[2] J.-P. Allouche and J.O. Shallit, The ring of k-regular sequences. Theoret. Comput. Sci. 98 (1992) 163-197. | Zbl

[3] J.-P. Allouche and J.O. Shallit, The ring of k-regular sequences, II. Theoret. Comput. Sci. 307 (2003) 3-29. | Zbl

[4] J.-P. Allouche and J.O. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press (2003). | Zbl | MR

[5] J. Berstel, Axel Thue's Papers on Repetitions in Words: a Translation. Number 20 in Publications du Laboratoire de Combinatoire et d'Informatique Mathématique. Université du Québec à Montréal (February 1995).

[6] S. Brlek, Enumeration of factors in the Thue-Morse word. Disc. Appl. Math. 24 (1989) 83-96. | Zbl

[7] J.J. Pansiot, The Morse sequence and iterated morphisms. Inform. Process. Lett. 12 (1981) 68-70. | Zbl

[8] H. Prodinger and F.J. Urbanek, Infinite 0-1-sequences without long adjacent identical blocks. Discrete Math. 28 (1979) 277-289. | Zbl

[9] A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 1 (1912) 1-67. Reprinted in Selected Mathematical Papers of Axel Thue, T. Nagell, editor, Universitetsforlaget, Oslo (1977) 413-478. | JFM

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