Farey sequence

The $n$’th Farey sequence is the ascending sequence of all rationals$\{0\le \frac{a}{b}\le 1:b\le n\}$.

The first 5 Farey sequences are

Farey sequences are a singularly useful tool in understanding the convergents that appear in continued fractions. The convergents for any irrational $\alpha$ can be found: they are precisely the closest number to $\alpha$ on the sequences $F_n$.

It is also of value to look at the sequences $F_n$ as $n$ grows. If $\frac{a}{b}$ and $\frac{c}{d}$ are reduced representations of adjacent terms in some Farey sequence $F_n$ (where $b,d\le n$), then they are adjacent fractions; their difference is the least possible:

$$\left|\frac{a}{b}-\frac{c}{d}\right|=\frac{1}{bd}.$$

Furthermore, the first fraction to appear between the two in a Farey sequence is $\frac{a+c}{b+d}$, in sequence $F_{b+d}$, and (as written here) this fraction is already reduced.

An alternate view of the “dynamics” of how Farey sequences develop is given by Stern-Brocot trees.