# Squares and overlaps in the Thue-Morse sequence and some variants

Shandy Brown; Narad Rampersad; Jeffrey Shallit; Troy Vasiga

RAIRO - Theoretical Informatics and Applications (2006)

- Volume: 40, Issue: 3, page 473-484
- ISSN: 0988-3754

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topBrown, Shandy, et al. "Squares and overlaps in the Thue-Morse sequence and some variants." RAIRO - Theoretical Informatics and Applications 40.3 (2006): 473-484. <http://eudml.org/doc/249704>.

@article{Brown2006,

abstract = {
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.
},

author = {Brown, Shandy, Rampersad, Narad, Shallit, Jeffrey, Vasiga, Troy},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {Thue-Morse word; overlap-free word; automatic sequence.; automatic sequence},

language = {eng},

month = {10},

number = {3},

pages = {473-484},

publisher = {EDP Sciences},

title = {Squares and overlaps in the Thue-Morse sequence and some variants},

url = {http://eudml.org/doc/249704},

volume = {40},

year = {2006},

}

TY - JOUR

AU - Brown, Shandy

AU - Rampersad, Narad

AU - Shallit, Jeffrey

AU - Vasiga, Troy

TI - Squares and overlaps in the Thue-Morse sequence and some variants

JO - RAIRO - Theoretical Informatics and Applications

DA - 2006/10//

PB - EDP Sciences

VL - 40

IS - 3

SP - 473

EP - 484

AB -
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.

LA - eng

KW - Thue-Morse word; overlap-free word; automatic sequence.; automatic sequence

UR - http://eudml.org/doc/249704

ER -

## References

top- J.-P. Allouche, J. Currie, and J. Shallit, Extremal infinite overlap-free binary words. Electronic J. Combinatorics5 (1998), #R27 (electronic). URIhttp://www.combinatorics.org/Volume_5/Abstracts/v5i1r27.html
- J.-P. Allouche and J.O. Shallit, The ring of k-regular sequences. Theoret. Comput. Sci.98 (1992) 163–197.
- J.-P. Allouche and J.O. Shallit, The ring of k-regular sequences, II. Theoret. Comput. Sci.307 (2003) 3–29.
- J.-P. Allouche and J.O. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press (2003).
- 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).
- S. Brlek, Enumeration of factors in the Thue-Morse word. Disc. Appl. Math.24 (1989) 83–96.
- J.J. Pansiot, The Morse sequence and iterated morphisms. Inform. Process. Lett.12 (1981) 68–70.
- H. Prodinger and F.J. Urbanek, Infinite 0–1-sequences without long adjacent identical blocks. Discrete Math.28 (1979) 277–289.
- 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.

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