Farey sequence

FCC link

The Farey sequence Fn of order n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size.

The Farey sequence is sometimes incorrectly called a Farey series.

Each Farey sequence:

• starts with the value 0, denoted by the fraction $ \frac{0}{1} $
• ends with the value 1, denoted by the fraction $ \frac{1}{1}$.

The Farey sequences of orders 1 to 5 are:

• ${\bf\it{F}}_1 = \frac{0}{1}, \frac{1}{1}$
• ${\bf\it{F}}_2 = \frac{0}{1}, \frac{1}{2}, \frac{1}{1}$
• ${\bf\it{F}}_3 = \frac{0}{1}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{1}{1}$
• ${\bf\it{F}}_4 = \frac{0}{1}, \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{1}{1}$
• ${\bf\it{F}}_5 = \frac{0}{1}, \frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{1}{1}$