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 divided into three parts. In Part 1, we define "points of symmetry," "intervals" and "generating fractions" and discuss general properties of the Farey sequence of Fibonacci numbers. In Part 2, we define conjugate fractions and deal with properties associated with intervals. Part 3 considers the Farey sequence of Fibonacci numbers as having been divided into stages and contains properties associated with "corresponding fractions" and "corresponding stages." A generalization of the Farey sequence of Fibonacci numbers is given at the end of the third part. The Farey sequence of Fibonacci numbers of order Fn (where Fn stands for the n term of the Fibonacci sequence) is the set of all possible fractions Fj/Fj, l=Q, 1, Z 3, —, n 1, j = 1, 2, 3, —, n (i <j) arranged in ascending order of magnitude. The last term is 1/1, i.e., Fj /F2. The first term is 0/Fn^j. We set FQ = O so that FQ +Ff = F2,F1 = F2=1. For convenience we denote a Farey sequence of Fibonacci numbers by f-f, that of order Fn by f-fn and the r term in the new Farey sequence of order Fn by f(r)n . PART 1 DEFINITION 1.1. Besides 1/1 we define an f(rjn to be a point of symmetry if f(r+i)n d f(r-Dn have ths same denominator. We have shown in an appendix the Farey sequence of all Fibonacci numbers up to 34. DEFINITION 1.2. We define an interval to be set of all f-fn fractions between two consecutive points of symmetry,. The interval may be closed or open depending upon the inclusion or omission of the points of symmetry. A closed interval is denoted by [ ] and an open interval by ( ) . DEFINITION 1.3. The distance between f(r)k and f(k)n is equal to \r-k\. Theorem 1.1. If f(r)n is a point of symmetry then it is of the form 1/F,% Moreover f(r+k)n ^ Ur-k)n n a v e the same denominator if they do not pass beyond the next point of symmetry on either side. The converse is also true. Proof. In the f-f sequence the terms are arranged in the following fashion. The terms in the last interval are of the form F/_? /Fj. The terms in the interval prior to that last are of the form Fj^/Fj —. If there are two fractions FM/FJ-1 and Fj_2/Fj-2 then their mediant* Fj/Fj lies in between them. That is,