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Abstract
Fibonacci numbers, Lucas numbers and Mulatu numbers are built in the same method. The three numbers differ in the first term, while the second term is entirely the same. The next terms are the sum of two successive terms. In this article, generalizations of Fibonacci, Lucas and Mulatu (GFLM) numbers are built which are generalizations of the three types of numbers. The Binet formula is then built for the GFLM numbers, and determines the golden ratio, silver ratio and Bronze ratio of the GFLM numbers. This article also presents generalizations of these three types of ratios, called Metallic ratios. In the last part we state the Metallic ratio in the form of continued fraction and nested radicals.
Highlights
Fibonacci numbers are found by Leonardo Fibonacci
The n th Fibonacci numbers is the sum of the numbers of the two previous terms where n ≥ 2 and the first and second terms are 0 and 1
The purpose of this study is to find a sequence of numbers which is a generalization of rows of Fibonacci, Lucas and Mulatu numbers
Summary
The n th Fibonacci numbers is the sum of the numbers of the two previous terms where n ≥ 2 and the first and second terms are 0 and 1. In addition to Fibonacci numbers, there are Lucas and Mulatu numbers where the n th term with n ≥ 2 is the sum of the numbers of the two previous terms, so that Lucas and Mulatu numbers have the same recursive function as Fibonacci numbers (Patel, D., and Lemma, 2011; Lemma et al, 2016; Lemma, 2019). Lucas's numbers is obtained by taking the two initial terms, the 0-th term is 2 and the 1st term is 1. Mulatu numbers is obtained by taking two initial terms, the 0-th term is 4 and the 1st term is 1
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