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