Farey sequence

A Farey sequence is sometimes called a Farey series, although the terms are not summed. The first eight Farey sequences are: Plotting the numerators versus the denominators of a Farey sequence gives a shape like the one to the right, shown for F 6 {\displaystyle F_{6}} . 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 2 n {\displaystyle 2n} , centered at the origin. By Pick's theorem, the area of the sunburst is 4 | F n | − 4 {\displaystyle 4|F_{n}|-4} , where | F n | {\displaystyle |F_{n}|} 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.