What is the most efficient way to determine the Farey sequence of degree n?

Question

I am going to implement a Farey fraction approximation for converting limited-precision user input into possibly-repeating rationals.
http://mathworld.wolfram.com/FareySequence.html

I can easily locate the closest Farey fraction in a sequence, and I can find Fn by recursively searching for mediant fractions by building the Stern-Brocot tree.
http://mathworld.wolfram.com/Stern-BrocotTree.html

However, the method I've come up with for finding the fractions in the sequence Fn seems very inefficient:
(pseudo)

For int i = 0 to fractions.count -2
{
    if fractions[i].denominator + fractions[i+1].denominator < n
    {    
        insert new fraction(
            numerator = fractions[i].numerator + fractions[i+1].numerator
            ,denominator = fractions[i].denominator + fractions[i+1].denominator)
            //note that fraction will reduce itself
        addedAnElement = true
    }
}
if addedAnElement 
    repeat

I will almost always be defining the sequence Fn where n = 10^m where m >1

So perhaps it might be best to build the sequence one time and cache it... but it still seems like there should be a better way to derive it.

EDIT:
This paper has a promising algorithm:
http://www.math.harvard.edu/~corina/publications/farey.pdf

I will try to implement.
The trouble is that their "most efficient" algorithm requires knowing the prior two elements. I know element one of any sequence is 1/n but finding the second element seems a challenge...

EDIT2:
I'm not sure how I overlooked this:
Given F0 = 1/n
If x > 2 then
F1 = 1/(n-1)

Therefore for all n > 2, the first two fractions will always be
1/n, 1/(n-1) and I can implement the solution from Patrascu.

So now, we the answer to this question should prove that this solution is or isn't optimal using benchmarks..

Answer 1

Neighboring fractions in Farey sequences are described in Sec. 3 of Neighboring Fractions in Farey Subsequences, http://arxiv.org/abs/0801.1981 .

Answer 2

Why do you need the Farey series at all? Using continued fractions would give you the same approximation online without precalculating the series.