The Farey sequence is most easily generated through the use of what is known as the Stern Berchot method.
This method centers around the incorrect method of adding fractions
that many grade school, high school, and even some college
(a/b) + (c/d) = (a+c)/(b+d)
Starting with the F1
case of 0/1 and 1/1, one simply adds two adjacent members of the sequence to generate the next level. An example is in order:
for n=1: 0/1, 1/1
n=2: 0/1, 0+1/1+1 (1/2), 1/1
n=3: 0/1, 0+1/1+2 (1/3), 1/2, 1+1/1+2 (2/3), 1/1
If you're going to generate the sequence in a program, it's best to use two list iterator
s, one behind the other, to generate the next level.