A Clojure fareysequences.clj namespace, and a Haskell FareySequencesModule.hs module, and a Python fareysequences.py module, exporting a dozen of functions coded after several selected formulas presented in the monograph A.O. Matveev, Farey Sequences: Duality and Maps Between Subsequences, Berlin, Boston: De Gruyter, 2017.
(See Table 1.1 of the monograph):
- (Standard) Farey Sequence Fm ( in LaTeX:
$\mathcal{F}_m$ ), where$m > 0$ ; - Farey Subsequence Fml ( in LaTeX:
$\mathcal{F}{}_m^l$ ), where$m > 1$ , and$0 < l < m$ ; - Farey Subsequence Gml ( in LaTeX:
$\mathcal{G}{}_m^l$ ), where$m > 1$ , and$0 < l < m$ ; - Farey Subsequence FB2mm ( in LaTeX:
$\mathcal{F}(\mathbb{B}(2m),m)$ ), where$m > 0$ ; - Farey Subsequence FBnm ( in LaTeX:
$\mathcal{F}(\mathbb{B}(n),m)$ ), where$n > 1$ , and$0 < m < n$ .
In healthy situations, all of the exported functions return reduced fractions h/k such that 0/1 <= h/k <= 1/1. If you get a negative fraction, it means that something went wrong, and the denominator of the resulting negative fraction has no computational meaning, since it just reports a reason of the problem---see the source code of the function you have used.
If you would like to build up a certain subsequence of neighboring (one after another) fractions in one of the above Personae, then first you should realize what you hold in your hands in the beginning.
If you have two neighboring fractions, then you proceed in the recurrent manner by means of a relatively fast function
of the form predecessor-of-pair-of-neighbors-in-personage or successor-of-pair-of-neighbors-in-personage, in Clojure
(predecessorOfPairOfNeighborsInPersonage or successorOfPairOfNeighborsInPersonage, in Haskell; predecessor_of_pair_of_neighbors_in_personage
or successor_of_pair_of_neighbors_in_personage, in Python), depending on which direction (descending or ascending) you take.
If you have one fraction then, in order to obtain your starting pair of neighboring fractions, you make one-time use of a relatively slow
function of the form predecessor-in-personage or successor-in-personage, in Clojure (predecessorInPersonage or successorInPersonage, in Haskell;
predecessor_in_personage or successor_in_personage, in Python), and then you proceed by calling, in the recurrent manner, the corresponding relatively fast
function of the form predecessor-of-pair-of-neighbors-in-personage or successor-of-pair-of-neighbors-in-personage
(predecessorOfPairOfNeighborsInPersonage or successorOfPairOfNeighborsInPersonage; predecessor_of_pair_of_neighbors_in_personage
or successor_of_pair_of_neighbors_in_personage).