Each Farey sequence starts with the value 0, denoted by the fraction 0⁄1, and ends with the value 1, denoted by the fraction 1⁄1 (although some authors omit these terms).
A Farey sequence is sometimes called a Farey series, which is not strictly correct, because the terms are not summed.
Farey sequences are named after the British geologist John Farey, Sr., whose letter about these sequences was published in the Philosophical Magazine in 1816. Farey conjectured that each new term in a Farey sequence expansion is the mediant of its neighbours — however, so far as is known, he did not prove this property. Farey's letter was read by Cauchy, who provided a proof in his Exercises de mathématique, and attributed this result to Farey. In fact, another mathematician, C. Haros, had published similar results in 1802 which were almost certainly not known either to Farey or to Cauchy. Thus it was a historical accident that linked Farey's name with these sequences.
From this, we can relate the lengths of Fn and Fn−1 using Euler's totient function φ(n) :-
Using the fact that |F1| = 2, we can derive an expression for the length of Fn :-
The asymptotic behaviour of |Fn| is :-
If a⁄b and c⁄d are neighbours in a Farey sequence, with a⁄b < c⁄d, then their difference c⁄d − a⁄b is equal to 1⁄bd. Since
this is equivalent to saying that
Thus 1⁄3 and 2⁄5 are neighbours in F5, and their difference is 1⁄15.
The converse is also true. If
for positive integers a,b,c and d with a < b and c < d then a⁄b and c⁄d will be neighbours in the Farey sequence of order max(b,d).
If p⁄q has neighbours a⁄b and c⁄d in some Farey sequence, with
then p⁄q is the mediant of a⁄b and c⁄d — in other words,
And if a⁄b and c⁄d are neighbours in a Farey sequence then the first term that appears between them as the order of the Farey sequence is increased is
which first appears in the Farey sequence of order b + d.
Thus the first term to appear between 1⁄3 and 2⁄5 is 3⁄8, which appears in F8.
The Stern-Brocot tree is a data structure showing how the sequence is built up from 0 (= 0⁄1) and 1 (= 1⁄1), by taking successive mediants.
Fractions that appear as neighbours in a Farey sequence have closely related continued fraction expansions. Every fraction has two continued fraction expansions - in one the final term is 1; in the other the final term is greater than 1. If p⁄q, which first appears in Farey sequence Fq, has continued fraction expansions
then the nearest neighbour of p⁄q in Fq (which will be its neighbour with the larger denominator) has a continued fraction expansion
and its other neighbour has a continued fraction expansion
Thus 3⁄8 has the two continued fraction expansions [0; 2, 1, 1, 1] and [0; 2, 1, 2], and its neighbours in F8 are 2⁄5, which can be expanded as [0; 2, 1, 1]; and 1⁄3, which can be expanded as [0; 2, 1].
For every fraction p/q (in its lowest terms) there is a Ford circle C[p/q], which is the circle with radius 1/(2q2) and centre at (p/q, 1/(2q2)). Two Ford circles for different fractions are either disjoint or they are tangent to one another - two Ford circles never intersect. If 0 < p/q < 1 then the Ford circles that are tangent to C[p/q] are precisely the Ford circles for fractions that are neighbours of p/q in some Farey sequence.
Thus C[2/5] is tangent to C[1/2], C[1/3], C[3/7], C[3/8] etc.
A surprisingly simple algorithm exists to generate the terms in either traditional order (ascending) or non-traditional order (descending):
Brute force searches for solutions to Diophantine equations in rationals can often take advantage of the Farey series (to search only reduced forms). The lines marked (*) can also be modified to include any two adjacent terms so as to generate terms only larger (or smaller) than a given term.