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 give the relationship between regular continued fractions and Lehner fractions, using a procedure known as insertion}. Starting from the regular continued fraction expansion of any real irrational x, when the maximal number of insertions is applied one obtains the Lehner fraction of x. Insertions (and singularizations) show how these (and other) continued fraction expansions are related. We also investigate the relation between Lehner fractions and the Farey expansion (also known as the full...