A Sturmian sequence related to the uniqueness conjecture for Markoff numbers

Sturmian sequences appear in the work of Markoff on approximations of real numbers and minima of quadratic functions. In particular, Christoffel words, or equivalently pairs of relatively prime nonnegative integers, parametrize the Markoff numbers. It was asked by Frobenius if this parametrization is injective. We answer this conjecture for a particular subclass of these numbers, and show that a special Sturmian sequence of irrational slope determines the order of the Markoff numbers in this subclass.