Farey Series

The Farey series F_n of order n is the increasing sequence of 
irreducible fractions in the range 0 to 1 whose denominators do 
not exceed n.  For example, the Farey series of order 5 is

    0    1    1    1    2    1    3    2    3    4    1
    -    -    -    -    -    -    -    -    -    -    -
    1    5    4    3    5    2    5    3    4    5    1

Symmetrical terms of a Farey series are complementary, e.g., 
(2/5) + (3/5) = 1.  Also, notice that each term in a Farey series
is the mediant of the two surrounding terms.  In other words, if 
a/b, c/d, and e/f are three consecutive terms, then we have

                      c      a + e
                      -  =   -----
                      d      b + f


Another basic property is that if a/b and c/d are consecutive terms
of a Farey sequence, then bc - ad = 1.