Some special values of vertices of trees on the suborbital graphs

In the present study, the action of a congruence subgroup of S L(2, Z) on ℚ^ is examined. From this action and its properties, vertices of paths of minimal length on the suborbital graph Fu,N give rise to some special sequence values, that are alternate sequences such as identity, Fibonacci and Lucas sequences. These types of vertices also give rise to special continued fractions, hence from recurrence relations for continued fractions, values of these vertices and values of special sequences were associated.