TWO ASPECTS OF A GENERALIZED FIBONACCI SEQUENCE

Johan Matheus Tuwankotta

Abstract


In this paper we study the so-called generalized Fibonacci sequence: $x_{n+2} = \alpha x_{n+1} + \beta x_n, n\in \mathbb{N}$.  We derive an open domain around the origin of the parameter space where the sequence converges to $0$.  The limiting behavior on the boundary of this domain are: convergence to a nontrivial limit, $k$-periodic ($k\in \mathbb{N}$), or quasi-periodic.  We use the ratio of two consecutive terms of the sequence to construct a rational approximation for algebraic numbers of the form: $\sqrt{r}, r\in \mathbb{Q}$.  Using a similar idea, we extend this to higher dimension to construct a rational approximation for  $\sqrt[3]{ a + b\sqrt{c}} +  \sqrt[3]{ a - b\sqrt{c}} + d$.

DOI : http://dx.doi.org/10.22342/jims.21.1.173.1-17


Keywords


Generalized Fibonacci sequence, convergence, discrete dynamical system, rational approximation

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References


Atkins, J., Geist, R., Fibonacci numbers and computer algorithms, College Math. J., 18, (1987), 328–337.

Cooper, C., Classroom capsules: Application of a generalized Fibonacci sequence, College Math. J., 15 (1984), 145–148.

Edson, M., Yayenie, O., A new generalization of Fibonacci Sequence & Extended Binet’s formula, Integers 9 (2009), pp. 639-654.

Kalman, D., Mena, R., The Fibonacci Numbers—Exposed, Mathematics Magazine, vol 76, nr 3, June 2003, pp. 167–181.

Katz, V. J., A History of Mathematics: an introduction 2nd ed., Addison-Wesley (1998), Reading, Massachusetts, etc.

Koshy, T., Fibonacci and Lucas Numbers with Applications, Wiley (2001), New York.

Li, H-C., Complete and reduced residue systems of second-order recurrences modulo p, Fibonacci Quart. 38 (2000), 272281.

Lucas, E., Th ́eorie des fonctions num ́eriques simplement p ́eriodiques, Amer. J. Math. 1

(1878), 184 240, 289321.

Sun, Z-H.; Sun, Z-W., Fibonacci numbers and Fermats last theorem, Acta Arith. 60 (1992), 371 388.

Tuwankotta, J.M., Contractive Sequence, Majalah Ilmiah Matematika dan Ilmu Pengetahuan

Alam: INTEGRAL, vol. 2 No.2, Oktober 1997, FMIPA Universitas Parahyangan.

Walton, J.E., Horadam, A.F., Some Aspects of Generalized Fibonacci Sequence, The Fibonacci Quaterly, 12 (3) (1974), pp. 241–250.

Wall, D. D., Fibonacci series modulo m, Amer. Math. Monthly 67 (1960), 525532.




DOI: https://doi.org/10.22342/jims.21.1.173.1-17

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