Farey Sequences

The Farey sequence $F_n$ lists every rational in $[0,1]$ with denominator at most $n$ , in lowest terms, in increasing order.

Examples:

\begin{aligned} F_1 &= \left\{\tfrac{0}{1},\,\tfrac{1}{1}\right\}, & F_2 &= \left\{\tfrac{0}{1},\,\tfrac{1}{2},\,\tfrac{1}{1}\right\}, & F_3 &= \left\{\tfrac{0}{1},\,\tfrac{1}{3},\,\tfrac{1}{2},\,\tfrac{2}{3},\,\tfrac{1}{1}\right\}. \end{aligned}

Two neighbouring terms of $F_n$ are adjacent. The mediant of $\frac{a}{b}$ and $\frac{c}{d}$ is $\frac{a+c}{b+d}$ .

Key facts proved in the problem set below:

Distinct reduced fractions with the same denominator cannot be adjacent.
The mediant always lies strictly between its parents.
Adjacent fractions satisfy $b+d>n$ and $bc-ad=1$ .
Neighbouring distances are bounded between $\frac{1}{n(n-1)}$ and $\frac{1}{n}$ .