A Note on Golden Ratio and Higher Order Fibonacci Sequences

Paolo Emilio Ricci^1,

¹Mathematics Department, International Telematic University UniNettuno, Roma, Italia

Cite this paper:
Paolo Emilio Ricci. A Note on Golden Ratio and Higher Order Fibonacci Sequences. Turkish Journal of Analysis and Number Theory. 2020; 8(1):1-5. doi: 10.12691/tjant-8-1-1

Abstract

The Lucas formula representing integer powers of the Golden ratio in terms of Fibonacci numbers is derived starting from a general result on matrix powers. The same technique is applied to the Tribonacci sequence, and the extension to higher-order Fibonacci sequences is also considered. It is shown that, by using classical results on matrix theory, the problem can be treated in a general and uniform method.

Keywords:
golden ratio Fibonacci sequence Tribonacci numbers matrix theory

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