On evolving chains of cube-free binary sequences

Jacek M. Kowalski and Andrzej Pękalski
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 3, Pages 79—94
DOI: 10.7546/nntdm.2021.27.3.79-94
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Authors and affiliations

Jacek M. Kowalski
Department of Physics, University of North Texas
Denton TX 76293, United States

Andrzej Pękalski
Institute of Theoretical Physics, University of Wrocław
50-203 Wrocław, Poland


Chains of concatenated finite binary words are considered, where each word, except possibly the very first one, is composed of alternating blocks of zeroes and ones with block lengths not exceeding two. These chains are formed following two evolution schemes. The first scheme is standard, where alternating blocks are visited at random. In the second approach, proposed by us in this paper, each subsequent word of the chain is uniquely determined by its immediate predecessor, being formed as a specifically inflated version of that word. Famous Kolakoski sequence is then just one, very special example of such deterministic evolution when one starts from its third element. We present heuristic arguments supported by simulations indicating that all such deterministic infinite chains should have the asymptotic density of digit 1 equal 1/2 and that the subsequent word lengths asymptotically scale with factor of 3/2 and hence the density of 1’s in subsequent finite words may also tend to 1/2.


  • Automatic sequences, Random and deterministic evolution of binary words,
    Algorithmic combinatorics, Kolakoski and related sequences, Scaling law

2020 Mathematics Subject Classification

  • 11B83, 11B85, 11R45, 11Y55, 65B10, 68R15


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Cite this paper

Kowalski, J. M., & Pękalski, A. (2021). On evolving chains of cube-free binary sequences. Notes on Number Theory and Discrete Mathematics, 27(3), 79-94, doi: 10.7546/nntdm.2021.27.3.79-94.

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