Generalizations of Fibonacci and Lucas sequences

Abstract

In this paper, we consider the Hecke groups $H(\sqrt{q}),~q\geq 5$ prime number, and we find an interesting number sequence whichis denoted by $d_n.$ For $q=5$, we get $d_{2n}=L_{2n+1}$ and $d_{2n+1}=\sqrt{%5}F_{2n+2}$ where $L_{2n+1}$ is $(2n+1)$th Lucas number and $F_{2n+2}$ is $(2n +2)th$ Fibonacci number. From this sequence, we obtain two new sequenceswhich are, in a sense, generalizations of Fibonacci and Lucas sequences.