A Necessary Condition for the Existence of the Asymptotic Density in Kolakoski Sequence

Abdallah Hammam^1,

¹Département de Mathématiques, Faculté des sciences, Université Moulay Ismaïl, 50020 Meknès, Morocco

Cite this paper:
Abdallah Hammam. A Necessary Condition for the Existence of the Asymptotic Density in Kolakoski Sequence. Turkish Journal of Analysis and Number Theory. 2018; 6(2):57-60. doi: 10.12691/tjant-6-2-5

Abstract

We investigate here the Oldenburger-Kolakoski sequence with K₁=1. In the first part, we give some expressions of the discrepancy function δ(n) representing the difference between 2s and 1s in K₁K₂…K_n. The discrepancy could be interpreted as a perturbation of a certain equilibrium. Our main result is a necessary and sufficient condition for the existence of the asymptotic density. In the last section, we present an algorithm to generate the sequence terms and a formula for the term K_n.

Keywords:
Kolakoski sequence recursive formula asymptotic density

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References:

[1]	Keane and C. Series (eds.), Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford University Press, 1991, pp. 35-70.
[2]	C.Kimberling,http://faculty.evansville.edu/ck6/integer/index.html.
[3]	N. J. A. Sloane, The On-line Encyclopedia of Integer Sequences, published electronically at http://oeis.org.