Does there exist an infinite number string without any 'refrain'? Let us consider an infinite or finite number string which consists of $0,1,2$. Then, let us call an adjacent pair of repeating number(s) ' **a refrain** '. For example, we have three refrains in the following string : $$01\overline{2}\ \overline{2}01202\overline{12}\ \overline{12}10\overline{201}\ \overline{201}02$$ **Question** : Does there exist an infinite number string without any refrain? **Motivation** : I've known that there exists an infinite number string which consists of $0,1,2,3$ without any refrain. This got me interested in the above expectation, but I'm facing difficutly. Can anyone help?

What you call "refrain" is called a _square_ in the literature of combinatorics on words. There are many square-free words on an alphabet of three letters. An example is the sequence

$$1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, \ldots$$

which can be obtained by starting with $1$ and then using the morphism $1\to 123$, $2\to 13$, $3\to 2$. (See <