I’m currently interested in cube-free infinite binary words. (See my Mathoverflow questions here and here for more information. Also see my previous blog posts here, here, and here for context.)

One very interesting cube-free infinite binary word is the Kolakoski word (also known as the Kolakoski sequence). The sequence is named after William Kolakoski, who introduced it in a problem published in the *American Mathematical Monthly* in 1965.

It seems that it was recently (last year?) discovered that the sequence was published earlier, in 1939, by Rufus Oldenburger in his paper Exponent trajectories in symbolic dynamics (*Transactions of the American Mathematical Society*, vol. 46, pp. 453-466).

One currently open problem (see, for example, problem 10 here) is to prove (or disprove) that the limiting frequency of each of the two characters exists and is equal to 1/2. Perhaps some insight can be found in Oldenburger’s paper?

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Dear,

A new paper : Some new formulas for Kolakoski sequence, at :

http://pubs.sciepub.com/tjant/4/3/1

thanks a lot.

Thank you very much for the link.