Farey sequence
In mathematics, the Farey sequence of order n is 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.
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.
The Farey sequences of orders 1 to 8 are :
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, without offering proof, that each new term in a Farey sequence expansion is the mediant of its neighbours. Farey's letter was read by Cauchy, who provided a proof in his Exercices de mathématique, and attributed this result to Farey. In fact, another mathematician, Charles Haros, had published similar results in 1802 which were not known either to Farey or to Cauchy. Thus it was a historical accident that linked Farey's name with these sequences. This is an example of Stigler's law of eponymy.
The Farey sequence of order n contains all of the members of the Farey sequences of lower orders. In particular Fn contains all of the members of Fn−1 and also contains an additional fraction for each number that is less than n and coprime to n. Thus F6 consists of F5 together with the fractions 1/6 and 5/6.
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