Continued fractions by the nearest even number

There exist many types (algorithms) of one-dimensional continued fractions. Two of them are most widespread algorithms, namely, the classical regular continued fractions and the continued fractions to the nearest integer ones. They both are based on the Euclid's algorithm.
Here we propose and investigate a new fundamental type of continued fractions: the continued fractions to the nearest even number. We demonstrate that continued fractions of this type possess the majority of important properties of classical continued fractions.