The Kolakoski sequence and related conjectures about orbits

Abstract

The Kolakoski sequence is the unique infinite sequence with values in $\{1, 2\}$ and first term twems $1, 2, \ldots$ which equals the sequence of run-lengths of itself, we call this $K(1, 2).$ We define $K(m, n)$ similarly for $m+n$ odd. A well-known open problem is that its limiting density is one-half. Indeed, not much is known about the Kolakoski sequence. The focus of this paper in on conjectures related to the Kolakoski sequence which are more discrete in nature. We conjecture that a certain doubly infinite family of finite sequences $E_{1, n}\left(1^{2^j}, 1^{2j}\right)$ has odd length for all $j>0$ and even $n>0.$ We define $cf(m, n, d)$ to be the "correlation frequency" or limiting probability that terms in $K(m, n)$ which are $d$ apart are equal. We conjecture that the sign of $cf(m, n, d) - 1/2$ is periodic mod $m+n.$ We also discuss extensive empirical evidence for these conjectures.