We present here a new approach to investigate the Kolakoski sequence Kol(a, b). In the first part of this paper, we give some general identities. In the second, we state our main result, which concerns the frequency of the letters in the case where a and b are odd. Finally, we give an algorithm to compute the term K n in the particular case of Kol(1, 3).

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.

We investigate here the Oldenburger-Kolakoski sequence ( ) 1 n n K ≥ with 1 1. K = In the first part, wegive some expressions of the discrepancy function ( ) n δ representing the difference between 2s and 1s in 1 2 ... . n K K K 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 .n K