Farey sequence

Etymology
Named after British geologist, whose letter about the sequences was published in the  in 1816.

Noun

 * 1)  For a given positive integer n, 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.
 * 2) * 2007, Jakub Pawlewicz, Order Statistics in the Farey Sequences in Sublinear Time, Lars Arge, Michael Hoffmann, Emo Welzl (editors), Algorithms - ESA 2007: 15th Annual European Symposium, Proceedings, Springer, 4698, page 218,
 * The Farey sequence of order $$n$$ (denoted $$\mathcal{F}_n$$) is the increasing sequence of all irreducible fractions from interval $$\textstyle [0,1]$$ with denominators less than or equal to $$\textstyle n$$. The Farey sequences have numerous interesting properties and they are well known in the number theory and in the combinatorics.
 * The Farey sequence of order $$n$$ (denoted $$\mathcal{F}_n$$) is the increasing sequence of all irreducible fractions from interval $$\textstyle [0,1]$$ with denominators less than or equal to $$\textstyle n$$. The Farey sequences have numerous interesting properties and they are well known in the number theory and in the combinatorics.

Usage notes

 * The sequence for given $$n$$ may be called the Farey sequence of order $$n$$, and is often denoted $$F_n$$.
 * The sequences are cumulative: each $$F_n$$ is contained in $$F_{n+1}$$. The added elements are those fractions $$\textstyle \frac m {n+1}$$ for which $$m$$ and $$n+1$$ are coprime.
 * The restriction that the fraction be in the range $$[0,1]$$ (i.e., $$0\le$$ numerator $$\le$$ denominator) is sometimes omitted.
 * With the restriction in place, every Farey sequence begins with $$\textstyle\frac 0 1$$ and ends with $$\textstyle\frac 1 1$$.