Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: http://www.sciepub.com/journal/tjant
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Turkish Journal of Analysis and Number Theory. 2019, 7(2), 37-40
DOI: 10.12691/tjant-7-2-2
Open AccessArticle

A Cogent Argument that Supports the Conjecture of Keane in Kolakoski Sequence A000002

Abdallah Hammam1,

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

Pub. Date: March 16, 2019

Cite this paper:
Abdallah Hammam. A Cogent Argument that Supports the Conjecture of Keane in Kolakoski Sequence A000002. Turkish Journal of Analysis and Number Theory. 2019; 7(2):37-40. doi: 10.12691/tjant-7-2-2

Abstract

The aim of our investigation is an attempt to answer two still unsolved questions about Kolakoski sequence (Kn)n≥1: Is there an explicit expression of the nth term Kn, and the second one, known as the conjecture of Keane, claims that the asymptotic density of twos, is In the first section of this paper, we present a new formula for Kn according to K1, K2, …Kp where In the second part, we define three sequences satisfying the condition UiVi=Wi, and using the fact that (Vi) increases at least exponentially while (Wi) does not, we conclude that (Ui) should converge to zero. Our argument is inductive but so strong to insure the validity of the conjecture in concern with density of twos.

Keywords:
Kolakoski sequence recursive formula asymptotic density.

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References:

[1]  N. J. A. Sloane, The On-line Encyclopedia of Integer Sequences, published electronicallyat http ://oeis.org.
 
[2]  A. Hammam, Some new Formulas for the Kolakoski Sequence A000002. Turkish Journal of Analysis and Number Theory. 2016; 4(3): 54-59.
 
[3]  O. Bordellès and B. Cloitre, Bounds for the Kolakoski Sequence, Journal of Integer Sequences, Vol. 14 (2011).
 
[4]  B. Steinsky, A recursive formula for the Kolakoski sequence, J. Integer Seq. 9 (2006), Article 06.3.7.
 
[5]  A. Hammam, Some Formulas for the Generalized Kolakoski Sequence Kol(a, b). Turkish Journal of Analysis and Number Theory. 2017; 5(4):139-142.