Fraenkel word

A Fraenkel word over n letters is defined recursively by

[math]\displaystyle{ \displaystyle{ \begin{align*} F_1 &= \mathbf{0}, \\ F_2 &= \mathbf{010}, \\ F_3 &= \mathbf{0102010}, \\ &\ \ \vdots \\ F_{n} &= F_{n-1}(\mathbf{n-1})F_{n-1}. \end{align*}} }[/math]

Open problems

Fraenkel's conjecture asserts that the only balanced infinite words (periodic or not) over n letters with letter densities pairwise distinct are eventually (letter reassignments of) infinite repetitions of [math]\displaystyle{ F_n. }[/math] In particular, it implies that the only balanced primitive circular words over at least 3 letters that have "step count vectors" with pairwise distinct components are Fraenkel words. The conjecture is known to be true for arity 3 to 7.