Farey sequence

Farey diagram to F9 represented with circular arcs. In the SVG image, hover over a curve to highlight it and its terms.
Farey diagram to F9.
Symmetrical pattern made by the denominators of the Farey sequence, F9.
Symmetrical pattern made by the denominators of the Farey sequence, F25.

In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which 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. (Some authors omit these terms.)

A Farey sequence is sometimes called a Farey series, although the terms are not summed.

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