Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T08:59:55.241Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  18 October 2006

Shandy Brown ,
Narad Rampersad ,
Jeffrey Shallit  and
Troy Vasiga
Shandy Brown
Affiliation:
Digital Thinkery, 199 Carter Avenue, Waterloo, Ontario N2J 3K5, Canada; [email protected]
Narad Rampersad
Affiliation:
School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada; [email protected]; [email protected]; [email protected]
Jeffrey Shallit
Affiliation:
School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada; [email protected]; [email protected]; [email protected]
Troy Vasiga
Affiliation:
School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada; [email protected]; [email protected]; [email protected]
Get access

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.

Keywords

Thue-Morse wordoverlap-free wordautomatic sequence.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 40 , Issue 3: Word Avoidability Complexity And Morphisms (WACAM) , July 2006 , pp. 473 - 484
Copyright
© EDP Sciences, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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
Allouche, J.-P. and Shallit, J.O., The ring of k-regular sequences. Theoret. Comput. Sci. 98 (1992) 163–197. CrossRef
Allouche, J.-P. and Shallit, J.O., The ring of k-regular sequences, II. Theoret. Comput. Sci. 307 (2003) 3–29. CrossRef
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).
Brlek, S., Enumeration of factors in the Thue-Morse word. Disc. Appl. Math. 24 (1989) 83–96. CrossRef
Pansiot, J.J., The Morse sequence and iterated morphisms. Inform. Process. Lett. 12 (1981) 68–70. CrossRef
Prodinger, H. and Urbanek, F.J., Infinite 0–1-sequences without long adjacent identical blocks. Discrete Math. 28 (1979) 277–289. CrossRef
Thue, A., 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.