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

Abstract

The aim of our investigation is an attempt to answer two still unsolved questions about Kolakoski sequence 1 () n n K ≥ : Is there an explicit expression of the n th term , n K and the second one, known as the conjecture of Keane, claims that the asymptotic density of twos, is 1. 2 In the first section of this paper, we present a new formula for n K according to 1 2 , ,... p K K K where 4 9 p n ≈. In the second part, we define three sequences satisfying the condition i i i U V W = , and using the fact that () i V increases at least exponentially while () i W does not, we conclude that () i U should converge to zero. Our argument is inductive but so strong to insure the validity of the conjecture in concern with density of twos.