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

Abdallah Hammam^1,

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

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 (K_n)_n≥1: Is there an explicit expression of the n^th term K_n, 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 K_n according to K₁, K₂, …K_p where In the second part, we define three sequences satisfying the condition U_iV_i=W_i, and using the fact that (V_i) increases at least exponentially while (W_i) does not, we conclude that (U_i) 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:

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[2]	A. Hammam, Some new Formulas for the Kolakoski Sequence A000002. Turkish Journal of Analysis and Number Theory. 2016; 4(3): 54-59.
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[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.