The ascending (finite) sequence of reduced fractions between and 1 with denominator <= n is known as Fn, the Farey sequence of order n.

Examples:

F1 = 0/1, 1/1
F2 = 0/1, 1/2, 1/1
F3 = 0/1, 1/3, 1/2, 2/3, 1/1
F4 = 0/1, 1/4, 1/3, 1/2, 2/3, 1/1
F5 = 0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1
F6 = 0/1, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 1/1
F7 = 0/1, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 1/1
...
Farey sequences show "simplest" decompositions of the interval [0,1]. They turn out to have many beautiful properties. Using them, the theory of continued fractions has a rather elegant formulation.