Let be a sequence with a finite number of terms equal to 1. The distance sequence of is defined as a sequence of the numbers of -couples of given distances. The paper investigates such pairs of sequences that a is different from and .
We extend the Davenport and Erdős construction of normal numbers to the case.
Using the natural extensions for the Rosen maps, we give an infinite-order-chain representation of the sequence of the incomplete quotients of the Rosen fractions. Together with the ergodic behaviour of a certain homogeneous random system with complete connections, this allows us to solve a variant of Gauss-Kuzmin problem for the above fraction expansion.