The Kolakoski word can be defined in a number of different ways, but probably the simplest is as the unique infinite word k over the alphabet {1, 2}, starting with 1, such that the sequence of lengths of blocks of consecutive identical symbols is equal to k itself.

The first few terms are 12211212212211211221211212211211212212211212212112. As you can see, the lengths of successive blocks are 1,2,2,1,1, etc., the sequence itself.

W. Kolakoski asked [Elementary Problem 5304, Amer. Math. Monthly 72 (1965), 674] if the symbols "1" and "2" occur with limiting frequency 1/2. This problem is still open. The best result is that if the frequency exists, then the frequence is between .499 and .501, and is due to Chvátal in a technical report.

The image shows a discussion of the problem at an Oberwolfach meeting, December 16 1988.

-- JeffreyShallit - 13 Jul 2011

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