farey 0.6.0

Generate Farey sequences and Farey ranges

farey

Generate Farey sequences and Farey ranges.

Overview

The Farey sequence F_n is the sequence of all completely reduced fractions between 0 and 1, with denominators ≤ n, in ascending order.

A Farey range applies a Farey sequence to subdivide an arbitrary interval into rational points. In other words, it is the (sorted) union of all the rational ranges with (the reciprocal of their) steps equal to the elements of the Farey sequence.

farey_sequence

Usage

from farey import farey_sequence

farey_sequence(1)
# [Fraction(0, 1), Fraction(1, 1)]

farey_sequence(5)
# [Fraction(0, 1), Fraction(1, 5), Fraction(1, 4), Fraction(1, 3),
#  Fraction(2, 5), Fraction(1, 2), Fraction(3, 5), Fraction(2, 3),
#  Fraction(3, 4), Fraction(4, 5), Fraction(1, 1)]

farey_range

Usage

from farey import farey_range
from fractions import Fraction

farey_range(0, 1, 3)
# [Fraction(0, 1), Fraction(1, 3), Fraction(1, 2), Fraction(2, 3), Fraction(1, 1)]

farey_range(0, 2, 4)
# [Fraction(0, 1), Fraction(1, 4), Fraction(1, 3), Fraction(1, 2),
#  Fraction(2, 3), Fraction(3, 4), Fraction(1, 1), Fraction(5, 4),
#  Fraction(4, 3), Fraction(3, 2), Fraction(5, 3), Fraction(7, 4),
#  Fraction(2, 1)]

# Reversed
farey_range(0, 1, -5)
# [Fraction(1, 1), Fraction(4, 5), ..., Fraction(0, 1)]

Parameters

Parameter	Default	Description
start	(required)	Left endpoint of the interval
end	(required)	Right endpoint of the interval
step	None	Controls the Farey order and direction (see below)

Step conventions

The step parameter controls which Farey order is used and whether the result is ascending or descending:

Step value	Order	Direction
n (positive int)	n	ascending
-n (negative int)	n	descending
1/n (float or Fraction(1, n))	n	ascending
-1/n (float or Fraction(-1, n))	n	descending
None	1	ascending

All equivalent forms produce the same result:

farey_range(0, 2, 3)
farey_range(0, 2, Fraction(1, 3))
farey_range(0, 2, 1/3)
# all return the same list

See also

This Python package has been inspired by the following Wolfram functions:

• FareySequence (built-in Wolfram Language function);
• FareyRange (resource function contributed by Jan Mangaldan).

License

MIT