Continued fractions by the nearest even number. The short type

In the preprint “Continued fractions by the nearest even number”, we propose a new type of one-dimensional continued fractions. It has many properties of the classic continued fractions: convergence, uniqueness of expansion, periodicity for quadratic irrationalities. But they converge very slow. Here we propose a fast (short) type of even continued fractions. The modi»ed algorithm allows to replace a sequence of similar parts of even continued fraction by one linear fractional transformation.