Editing Farey sequence (section)

===Farey neighbours and continued fractions===
Fractions that appear as neighbours in a Farey sequence have closely related [[continued fraction]] expansions. Every fraction has two continued fraction expansions &mdash; in one the final term is 1; in the other the final term is greater by 1. If {{math|{{sfrac|''p''|''q''}}}}, which first appears in Farey sequence {{mvar|F<sub>q</sub>}}, has the [[simple continued fraction|continued fraction expansion]]s
<math display=block>\begin{align}
  &[0;\ a_1,\ a_2,\ \ldots,\ a_{n-1},\ a_n,\ 1] \\{}
  &[0;\ a_1,\ a_2,\ \ldots,\ a_{n-1},\ a_n + 1]
\end{align}</math>

then the nearest neighbour of {{math|{{sfrac|''p''|''q''}}}} in {{mvar|F{{sub|q}}}} (which will be its neighbour with the larger denominator) has a continued fraction expansion
<math display=block>[0;\ a_1,\ a_2,\ \ldots,\ a_n]</math>

and its other neighbour has a continued fraction expansion 
<math display=block>[0;\ a_1,\ a_2,\ \ldots,\ a_{n-1}]</math>

For example, {{sfrac|3|8}} has the two continued fraction expansions {{math|[0; 2, 1, 1, 1]}} and {{math|[0; 2, 1, 2]}}, and its neighbours in {{math|''F''<sub>8</sub>}} are {{sfrac|2|5}}, which can be expanded as {{math|[0; 2, 1, 1]}}; and {{sfrac|1|3}}, which can be expanded as {{math|[0; 2, 1]}}.