Generalizations of Fibonacci and Lucas sequences

Abstract

In this paper, we consider the Hecke groups H(√q), q ≥ 5 prime number, and we find an interesting number sequence which is denoted by dn. For q = 5, we get d2n = L2n+1 and d2n+1 = √5F2n+2 where L2n+1 is (2n + 1)th Lucas number and F2n+2 is (2n + 2)th Fibonacci number. From this sequence, we obtain two new sequences which are, in a sense, generalizations of Fibonacci and Lucas sequences.