The

ascending (

finite)

sequence of

reduced fractions between

and

1 with

denominator <=

`n` is known as

`F`_{n}, the

*Farey sequence* of order

`n`.

`F`_{1} = 0/1, 1/1

`F`_{2} = 0/1, 1/2, 1/1

`F`_{3} = 0/1, 1/3, 1/2, 2/3, 1/1

`F`_{4} = 0/1, 1/4, 1/3, 1/2, 2/3, 1/1

`F`_{5} = 0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1

`F`_{6} = 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

`F`_{7} = 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.