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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; shandy@geeky.net
Narad Rampersad
Affiliation:
School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada; nrampersad@math.uwaterloo.ca; shallit@graceland.uwaterloo.ca; tmjvasiga@cs.uwaterloo.ca
Jeffrey Shallit
Affiliation:
School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada; nrampersad@math.uwaterloo.ca; shallit@graceland.uwaterloo.ca; tmjvasiga@cs.uwaterloo.ca
Troy Vasiga
Affiliation:
School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada; nrampersad@math.uwaterloo.ca; shallit@graceland.uwaterloo.ca; tmjvasiga@cs.uwaterloo.ca
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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

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References

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