The author studies the Hecke group \(H(\sqrt q)\) (\(q\) a prime \(\geq 5\)), the subgroup of \(\text{PSL}(2, \mathbb{Z})\) generated by \(z \to {- {1/z}}\) and \(z \to {z + \sqrt q}\). This group can also be generated by \(z \to {- {1/z}}\) and an element \(S\) whose matrix representation is \[ \left(\begin{matrix} 0 & {-1} \\ 1 & {\sqrt q} \end{matrix}\right). \] The author proves that \(S^n\) can be computed from a single sequence \((d_n)_{n \geq 0}\) that resemble Fibonacci and Lucas sequences. Note that Reference [9] has appeared, see Zbl 1059.11035 (with almost the same title and co-author).