Symmetry in the sequence of approximation coefficients

Abstract:

Let $\{a_n\}_1^\infty$ and $\{\theta _n\}_0^\infty$ be the sequences of partial quotients and approximation coefficients for the continued fraction expansion of an irrational number. We will provide a function $f$ such that $a_{n+1} = f(\theta _{n\pm 1},\theta _n)$. In tandem with a formula due to Dajani and Kraaikamp, we will write $\theta _{n \pm 1}$ as a function of $(\theta _{n \mp 1}, \theta _n)$, revealing an elegant symmetry in this classical sequence and allowing for its recovery from a pair of consecutive terms.