@article{tjant2019722,
author={Hammam, Abdallah},
title={A Cogent Argument that Supports the Conjecture of Keane in Kolakoski Sequence A000002},
journal={Turkish Journal of Analysis and Number Theory},
volume={7},
number={2},
pages={37--40},
year={2019},
url={http://pubs.sciepub.com/tjant/7/2/2},
issn={2333-1232},
abstract={The aim of our investigation is an attempt to answer two still unsolved questions about Kolakoski sequence (K<SUB>n</SUB>)<SUB>n¡Ý1</SUB>: Is there an explicit expression of the <i>n</i><SUP>th</SUP> term K<SUB>n</SUB>, and the second one, known as the conjecture of Keane, claims that the asymptotic density of twos, is <img src=image/abs1.png></img> In the first section of this paper, we present a new formula for K<SUB>n</SUB> according to K<SUB>1</SUB>, K<SUB>2</SUB>, ¡­K<SUB>p</SUB> where <img src=image/abs2.png></img> In the second part, we define three sequences satisfying the condition U<SUB>i</SUB>V<SUB>i</SUB>=W<SUB>i</SUB>, and using the fact that (V<SUB>i</SUB>) increases at least exponentially while (W<SUB>i</SUB>) does not, we conclude that (U<SUB>i</SUB>) should converge to zero. Our argument is inductive but so strong to insure the validity of the conjecture in concern with density of twos.},
doi={10.12691/tjant-7-2-2}
publisher={Science and Education Publishing}
}