  Recently it has been found that some special subsequences within a Farey
sequence play a crucial role in determining the ranges of coupling constant for
which quantum soliton states can exist for an integrable derivative nonlinear
Schrodinger model. In this article, we find a novel mapping which connects two
such subsequences belonging to Farey sequences of different orders. By using
this mapping, we construct an algorithm to generate all of these special
subsequences within a Farey sequence. We also derive the continued fraction
expansions for all the elements belonging to a subsequence and observe a close
connection amongst the corresponding expansion coefficients.
