Farey sequence

The Farey sequence of order $n$, denoted $F_n$, is the ascending sequence of completely reduced fractions between $0$ and $1$ whose denominators do not exceed $n$. Each fraction $a/b$ in $F_n$ satisfies $0 \\leq a \\leq b \\leq n$ and $\gcd(a, b) = 1$.

Important properties

• Each Farey sequence starts with $0/1$ and ends with $1/1$.

• No two consecutive fractions in a Farey sequence are equal.

• If $a/b$ and $c/d$ are consecutive terms in $F_n$, then $bc - ad = 1$.

• Every Farey sequence $F_n$ contains all the terms of $F_{n-1}$, plus possibly some new fractions.