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.