More on the Fascinating Characterizations of Mulatu's Numbers

The discoveries of Mulatu's numbers, better known as Mulatu's sequence, represent revolutionary contributions to the mathematical world. The primary contribution of Mulatu numbers to the mathematical world is their introduction as a new, distinct, and highly related sequence to the well-established Fibonacci and Lucas numbers, providing a novel context for exploring number theory, recursive sequences, and their identities. His best-known work is Mulatu's sequence, in which each new number is the sum of the two preceding numbers. Previously proven results on $F_n,L_n$ and $M_n$ (Fibonacci's, Lucas's and Mulatu's numbers respectively) which create a motivation for this work are: equivalencies among ( $M_n,F_{n-3}+F_{n-1}+F_{n+2},L_n+2F_{n-1}$ , $F_n+4F_{n-1},4F_{n+1}-3F_n$ ); $L_n+2F_{2n-1}=M_n{L}_n+5{F_n}^2-{L_n}^2;M_{2n}=M_n{L}_n+5{F_n}^2-{L_n}^2;M_{2n}=M_n{L}_n+{4(-1)}^{n+1}$ . When various operations and manipulations are performed on the numbers in this sequence, remarkable and intricate patterns begin to emerge. This study aimed to identify novel characterizations of Mulatu's numbers.

This study employed a multi-faceted approach to investigate characterizations of Mulatu's numbers. Mathematical proof techniques such as principle of mathematical induction, proof by contradiction and direct proof were utilized to substantiate findings.

In this study, we provided several characterizations of Mulatu's numbers. We also investigated the properties and patterns of these numbers. Moreover, we have also shown that, similar to Fibonacci's numbers, Mulatu's numbers give the so-called golden ratio, which is most applicable in numerical optimization. Furthermore, we formulated a relation among Mulatu's numbers, Fibonacci numbers, and Lucas numbers. Finally, we provided a generating function for the Mulatu numbers.

In this study, we uncovered novel characterizations of Mulatu's numbers and introduced a generating function for them. We investigated relationship between Mulatu's numbers and the golden ratio. The results discussed offer valuable insights and enhance our understanding of their properties. Furthermore, these findings play a vital role in the boarder context of mathematics and related areas, contributing significantly to the field.