A family of Beta-Fibonacci sequences

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					  Scientia Magna
  Vol. 5 (2009), No. 3, 6-12




           A family of Beta-Fibonacci sequences
                                  Krongtong Ratanavongsawad
            Department of Mathematics Kasetsart University, Bangkok, Thailand
                                E-mail: fsciktr@ku.ac.th


    Abstract This paper gives a generalization of Beta-nacci and Fibonacci sequences and the
    general solution obtained is given in terms of Beta-Fibonacci numbers.

    Keywords      Beta-nacci sequence, Fibonacci sequence.


§1. Preliminaries and introduction
    The Fibonacci sequence, say {Fn }∞ is defined recurrently by
                                     n=0


                               Fn = Fn−1 + Fn−2 ,        for all n ≥ 2,                        (1)

with initial conditions


                                     F0 = 1;      F1 = 1.


The general term of the Fibonacci sequence is
                                      √     n+1                √          n+1
                            1      1+ 5             1       1− 5
                      Fn = √                      −√                            .
                             5       2               5        2

The Fibonacci sequence has been studied extensively and generalized in many ways. In [1]
Peter R. J. Asveld studied the class of recurrence relations

                                                             k
                                 Gn = Gn−1 + Gn−2 +               αj nj                        (2)
                                                            j=0

with initial conditions

                                     G0 = 1;      G1 = 1.

    The main result of [1] consists of an expression for Gn in terms of the Fibonacci numbers
Fn and Fn−1 , and in the parameters α0 , α1 , ..., αk .
    The Beta-nacci sequence, say {Bn }∞ is defined recurrently by
                                        n=0


                            Bn = Bn−1 + 2Bn−2 ,             for all n ≥ 2,                     (3)
Vol. 5                           A family of Beta-Fibonacci sequences                          7

with initial conditions

                                      B0 = 1;     B1 = 1.

     The general term of the Beta-nacci sequence is

				
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