Neo Balancing Numbers and their Role in Sequences

Abstract

This dissertation focuses on the study of neo balancing numbers and their relationships to classical balancing numbers, t-balancing numbers, and sequence t-balancing numbers. We introduce the concept of t-neo balancing numbers and derive their fundamental properties, including generating functions and recurrence relations. Furthermore, we explore the generalization of neo balancing numbers within sequences defined by linear expressions, and establish several significant results linking neo balancing numbers to Diophantine equations and Pell-type structures via important Brahmagupta's identity. Thus, we discover generalized sequence neo balancing numbers and their theorem. Furthermore, we also study certain sequences on sequence t-neo balancing number which can be expanded in the future. The findings contribute to the deeper understanding in number theory and open new directions for future research.

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