Mulatu Polynomials and an Efficient Detection Algorithm for Mulatu Numbers

Background Linear recurrence sequences have been extensively studied in number theory and combinatory, with the Fibonacci sequence being the most classical example. Recent research has expanded to include various generalizations such as k-Fibonacci sequences 1 Cullen sequences 2 and polynomial extensions [see 3,4 ]. Among these, Mulatu numbers, introduced by Mulatu Lemma 5 and defined by the recurrence: Mn=Mn−1+Mn−2,M0=4, M1=1, have emerged as an interesting variant with unique arithmetic properties. Recent work by Derso and Admasu 6 established several characterizations of Mulatu numbers, including sum formulas, divisibility properties, and connections to the golden ratio. Methods we develop and analyze an efficient detection algorithm for determining whether a given integer belongs to the Mulatu sequence, based on a perfect-square criterion and modular arithmetic. Our results unify and extend recent work on generalized by Fibonacci sequences and Lucas Sequences and provide new computational tools for number theory and discrete mathematics. Results We derive explicit Binet-type formulas, generating functions, and combinatorial identities, establishing deep connections with Fibonacci polynomials, Lucas polynomials, and other linear recurrence sequences. Conclusions This paper gives a polynomial generalization of Mulatu numbers that extends the classical recurrence Mn=Mn−1+Mn−2 to polynomial sequences.