On Farey sequence and quadratic Farey sums

Computing Order Statistics in the Farey Sequence

We study the problem of computing the k-th term of the Farey sequence of order n, for given n and k. Several methods for generating the entire Farey sequence are known. However, these algorithms require at least quadratic time, since the Farey sequence has Θ(n) elements. For the problem of finding the k-th element, we obtain an algorithm that runs in time O(n lgn) and uses space O( √ n). The sa...

A Farey Sequence of Fibonacci Numbers

The Farey sequence is an old and famous set of fractions associated with the integers. We here show that if we form a Farey sequence of Fibonacci Numbers, the properties of the Farey sequence are remarkably preserved (see [2]). In fact we find that with the new sequence we are able to observe and identify "points of symmetry/' "intervals," "generating fractions" and "stages." The paper is divid...

Farey Curves

^nr", Hn , INTRODUCTION 4. Observations and Questions References Consider the family of quadratic polynomials PA : C —> C, where A is a complex number with |A| < 1, defined by Px(z) = Xz + z. We will examine this polynomial from a dynamical point of view; that is, we will be interested in its behaviour under iteration. The first observation is that the polynomial Px has a fixed point at zero: P...

Products of Farey Fractions

On the Index of Farey Sequences

For any two consecutive Farey fractions γi = ai/qi < γi+1 = ai+1/qi+1, one has ai+1qi − aiqi+1 = 1 and qi + qi+1 > Q. Conversely, if q and q ′ are two coprime integers in {1, . . . , Q} with q + q > Q, then there are unique a ∈ {1, . . . , q} and a ∈ {1, . . . , q} for which aq − aq = 1, and a/q < a/q are consecutive Farey fractions of order Q. Therefore, the pairs of coprime integers (q, q) wi...