On Generalization of Fibonacci, Lucas and Mulatu Numbers

Agung Prabowo (Jurusan Matematika Fakultas Matematika dan Ilmu Pengetahuan Alam Universitas Jenderal Soedirman)

International Journal of Quantitative Research and Modeling
Vol 1, No 3 (2020)

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.