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The Linus Sequence

Published online by Cambridge University Press:  22 June 2009

PAUL BALISTER ,
STEVE KALIKOW  and
AMITES SARKAR
PAUL BALISTER
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA (e-mail: pbalistr@memphis.edu, skalikow@memphis.edu)
STEVE KALIKOW
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA (e-mail: pbalistr@memphis.edu, skalikow@memphis.edu)
AMITES SARKAR
Affiliation:
Department of Mathematics, Western Washington University, Bellingham, WA 98225, USA (e-mail: amites.sarkar@wwu.edu)
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Abstract

Define the Linus sequence Ln for n ≥ 1 as a 0–1 sequence with L1 = 0, and Ln chosen so as to minimize the length of the longest immediately repeated block Ln−2r+1 ⋅⋅⋅ Ln−r = Ln−r+1 ⋅⋅⋅ Ln. Define the Sally sequence Sn as the length r of the longest repeated block that was avoided by the choice of Ln. We prove several results about these sequences, such as exponential decay of the frequency of highly periodic subwords of the Linus sequence, zero entropy of any stationary process obtained as a limit of word frequencies in the Linus sequence and infinite average value of the Sally sequence. In addition we make a number of conjectures about both sequences.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 1 , January 2010 , pp. 21 - 46
Copyright
Copyright © Cambridge University Press 2009

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