On Generalization of Fibonacci, Lucas and Mulatu Numbers

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.


Introduction
Fibonacci numbers are found by Leonardo Fibonacci. Fibonacci was born in Pisa in 1170 (Koshy, 2001;Kalman, and Mena, 2003;Falcón and Plaza, 2007). The th n  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 th n  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 was found by Francois Edouard Anatole Lucas (Koshy, 2001). 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 were discovered by Lemma Mulatu andpublished in 2011 (Mulatu, 2016). Mulatu numbers is obtained by taking two initial terms, the 0-th term is 4 and the 1st term is 1.
Interesting properties of Lucas's numbers have been reviewed by Lemma (2011). Schneider (2016) discusses the golden ratio in the form of continued fraction and nested radicals. Whereas Sivaraman (2020) developed Metallic ratio which is a generalization of three types of ratios namely Golden, Silver and Bronze ratios.
Similarities in the formation of Fibonacci numbers, Lucas and Mulatu numbers produce ideas to obtain new numbers which are generalizations of the three. The resulting generalization also produces other e-ISSN 2721-477X p-ISSN 2722 numbers that have not yet been found. Interesting properties of the new numbers are examined in this article and the results are obtained that these properties have been attached to Fibonacci, Lucas and Mulatu numbers.

Research Methodology
The purpose of this study is to find a sequence of numbers which is a generalization of rows of Fibonacci, Lucas and Mulatu numbers. Interesting properties of the new sequence numbers are examined in the article. The steps taken to achieve these two objectives are: 1. define a new sequence of numbers which is a generalization of Fibonacci Lucas and Mulatu numbers. This new sequence of numbers is called Generalization of Fibonacci-Lucas-Mulatu (GFLM) numbers; 2. shows that the solution of the GFLM numbers recursive relation is 3. build the Binet formula for GFLM numbers and show that the formula produces the Binet formula for Fibonacci, Lucas and Mulatu numbers; 4. shows that the ratio of two consecutive terms in the GFLM number converges to the golden ratio; establishing a silver ratio and a bronze ratio for the GFLM numbers; 5. build a Metallic ratio of order for GFLM numbers. This section also discusses several special cases related to Metallic ratio; and 6. states the Metallic ratio in the form of continued fractions and nested radicals.

Homogeneous Linear Recursive Relations with Constant Coefficients (HLRRCC) is equation
with all terms in a one-rank recursive relation, not multiplication of several terms, and the coefficient of all terms is a constant. HLRRCC with order th k  given in Equation ( (4) n a : th n  term;   , : nonzero real constant.
The solution of Equation (4) is sought by assuming 2 r a n  so we get Equation (5) Real roots of quadratic equation (5) The recursive relation of Equation (5) can be formed into the general solution of the recursive relation. Proof of a general solution of the recursive relation can be seen in Theorem 1.

Theorem 1.
Suppose that 1 r and 2 r the solution differs from the equation has a solution from its has the solution of the recursive relation is . Theorem 1 can be proved by means of an equation With elimination method, the solution of Equations (8) and (9)

Golden Ratio on the GFLM Numbers
For the GFLM numbers (2) The solutions of Equation (17)  can be stated by Equation (18)  .
. This result is used to prove theorem 2.
Theorem 2. The ratio of two consecutive terms in the GFLM number converges to the golden ratio, viz In the next section we will build a new line that is raised from the GFLM number. From these new numbers, two ratios are called the Silver ratio and the Bronze ratio. In the next section, a Metallic ratio is built which is a generalization of the ratio of gold, silver and bronze.

Silver Ratio on the GFLM Numbers
Consider at the sequence of numbers in (20) Table 2. The ratio of two consecutive numbers for Table 2 converges to 2.4142.  (20) converging to silver ratio. Because

Bronze Ratio on the GFLM Numbers
Consider at the sequence of numbers in (23) Table 3. The ratio of two consecutive numbers to 2 1, 0, k  in Table 3 converge towards 3.3028.

Metallic Ratio on the GFLM Numbers
We have obtained three types of ratios namely the ratio of Gold, Silver and Bronze with successive values is will be faster towards 1 for r the greater one. So, the Rhodium ratio is faster to 1 than the platinum ratio. For this case, we get the result that the Platinum ratio and Rhodium ratio are the same.

The Infinite Big Order
For t large, Metallic ratio will increase. We get, for