Faye tackled this problem :
$F_1$: $\frac{0}{1}$, $\frac{1}{1}$
$F_2$: $\frac{0}{1}$, $\frac{1}{2}$, $\frac{1}{1}$
$F_3$: $\frac{0}{1}$, $\frac{1}{3}$, $\frac{1}{2}$, $\frac{2}{3}$, $\frac{1}{1}$
$F_4$: $\frac{0}{1}$, $\frac{1}{4}$,$\frac{1}{3}$, $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$, $\frac{1}{1}$
$F_5$: $\frac{0}{1}$, $\frac{1}{5}$, $\frac{1}{4}$,$\frac{1}{3}$, $\frac{2}{5}$, $\frac{1}{2}$, $\frac{3}{5}$, $\frac{2}{3}$, $\frac{3}{4}$, $\frac{4}{5}$, $\frac{1}{1}$
$F_6$: $\frac{0}{1}$, $\frac{1}{6}$, $\frac{1}{5}$, $\frac{1}{4}$, $\frac{1}{3}$, $\frac{2}{5}$, $\frac{1}{2}$, $\frac{3}{5}$, $\frac{2}{3}$, $\frac{3}{4}$, $\frac{4}{5}$, $\frac{5}{6}$, $\frac{1}{1}$
$F_7$: $\frac{0}{1}$, $\frac{1}{7}$, $\frac{1}{6}$, $\frac{1}{5}$, $\frac{1}{4}$, $\frac{2}{7}$, $\frac{1}{3}$, $\frac{2}{5}$, $\frac{3}{7}$, $\frac{1}{2}$, $\frac{4}{7}$, $\frac{3}{5}$, $\frac{2}{3}$, $\frac{5}{7}$, $\frac{3}{4}$, $\frac{4}{5}$, $\frac{5}{6}$, $\frac{6}{7}$, $\frac{1}{1}$
$F_8$: $\frac{0}{1}$, $\frac{1}{8}$, $\frac{1}{7}$, $\frac{1}{6}$, $\frac{1}{5}$, $\frac{1}{4}$, $\frac{2}{7}$, $\frac{1}{3}$, $\frac{3}{8}$, $\frac{2}{5}$, $\frac{3}{7}$, $\frac{1}{2}$, $\frac{4}{7}$, $\frac{3}{5}$, $\frac{5}{8}$, $\frac{2}{3}$, $\frac{5}{7}$, $\frac{3}{4}$, $\frac{4}{5}$, $\frac{5}{6}$, $\frac{6}{7}$, $\frac{7}{8}$, $\frac{1}{1}$
$F_9$: $\frac{0}{1}$, $\frac{1}{9}$, $\frac{1}{8}$, $\frac{1}{7}$, $\frac{1}{6}$, $\frac{1}{5}$, $\frac{2}{9}$, $\frac{1}{4}$, $\frac{2}{7}$, $\frac{1}{3}$, $\frac{3}{8}$, $\frac{2}{5}$, $\frac{3}{7}$, $\frac{4}{9}$, $\frac{1}{2}$, $\frac{5}{9}$, $\frac{4}{7}$, $\frac{3}{5}$, $\frac{5}{8}$, $\frac{2}{3}$, $\frac{5}{7}$, $\frac{3}{4}$, $\frac{7}{9}$, $\frac{4}{5}$, $\frac{5}{6}$, $\frac{6}{7}$, $\frac{7}{8}$, $\frac{8}{9}$, $\frac{1}{1}$
$F_{10}$:$\frac{0}{1}$, $\frac{1}{10}$, $\frac{1}{9}$, $\frac{1}{8}$, $\frac{1}{7}$, $\frac{1}{6}$, $\frac{1}{5}$, $\frac{2}{9}$, $\frac{1}{4}$, $\frac{2}{7}$, $\frac{3}{10}$, $\frac{1}{3}$, $\frac{3}{8}$, $\frac{2}{5}$, $\frac{3}{7}$, $\frac{4}{9}$, $\frac{1}{2}$, $\frac{5}{9}$, $\frac{4}{7}$, $\frac{3}{5}$, $\frac{5}{8}$, $\frac{2}{3}$, $\frac{7}{10}$, $\frac{5}{7}$, $\frac{3}{4}$, $\frac{7}{9}$, $\frac{4}{5}$, $\frac{5}{6}$, $\frac{6}{7}$, $\frac{7}{8}$, $\frac{8}{9}$, $\frac{9}{10}$, $\frac{1}{1}$.

The sequence you get by putting in those mediants is the fifth Farey sequence. When I tried it with the others, I discovered that if you're doing the nth Farey sequence and you put in the mediants that will give you n+1 on the denominator, you get the n+1th Farey sequence.