Generalizations of Fibonacci and Lucas sequences


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.

DOI Code: 10.1285/i15900932v21n1p113

Keywords: Hecke groups; Fibonacci numbers; Lucas numbers

Classification: 11B39; 20H10

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