Farey sequence - Wikipedia
In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction,[ a] which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size.
With the restricted definition, 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.[ 2]
Reflecting this shape around the diagonal and main axes generates the Farey sunburst, shown below. The Farey sunburst of order n connects the visible integer grid points from the origin in the square of side 2n , centered at the origin. Using Pick's theorem, the area of the sunburst is 4(|Fn | − 1) , where |Fn | is the number of fractions in Fn .
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.[ 4] Thus it was a historical accident that linked Farey's name with these sequences. This is an example of Stigler's law of eponymy.
