A Necessary Condition for the Existence of the Asymptotic Density in Kolakoski Sequence

Abstract

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