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Characterization of two-distance sequences

Part of: Sequences and sets Designs and configurations Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  09 April 2009

W. F. Lunnon  and
P. A. B. Pleasants
W. F. Lunnon
Affiliation:
Department of Computer ScienceSt. Patrick's CollegeMaynoothCounty KildareEire
P. A. B. Pleasants
Affiliation:
School of Mathematics, Physics Computing and ElectronicsMacQuarie UniversityNSW 2109, Australia
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Abstract

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Three differently defined classes of two-symbol sequences, which we call the two-distance sequences, the linear sequences and the characteristic sequences, have been discussed by a number of authors and some equivalences between them are known. We present a self-contained proof that the three classes are the same (when ambiguous cases of linear sequences are suitably in terpreted). Associated with each sequence is a real invariant (having a different appropriate definition for each of the three classes). We give results on the relation between sequences with the same invariant and on the symmetry of the sequences. The sequences are closely related to Beatty sequences and occur as digitized straight lines and quasicrystals. They also provide examples of minimal word proliferation in formal languages.

Keywords

Beatty sequencesFibonacci sequencessymbolic dynamicschain codesdigitized straight linesquasicrystals.

MSC classification

Secondary: 11B99: None of the above, but in this section 05B99: None of the above, but in this section 68R05: Combinatorics 68U05: Computer graphics; computational geometry
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 53 , Issue 2 , October 1992 , pp. 198 - 218
Copyright
Copyright © Australian Mathematical Society 1992

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