Symmetries in the sequence of approximation coefficients

Let {an}1 and {θn}0 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 an+1 = f (θn±1,θn). In tandem with a formula due to Dajani and Kraaikamp, we will write θn±1 as a function of (θn∓1,θn), revealing an elegant symmetry in this classical sequence and allowing for its recovery from a pair of consecutive terms.