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

For circular words (equivalently, infinite periodic words), Fraenkel's conjecture asserts that the only balanced circular words over n ≥ 3 letters with letter occurrences pairwise distinct are (letter reassignments of) [math]\displaystyle{ F_n. }[/math] The conjecture is known to be true for 3 ≤ n ≤ 7.