Generalization of properties of fibonacci- like number sequences

In this study, special number sequences, namely the Fibonacci, Lucas, and Gibonacci sequences, are examined from a theoretical perspective. Initially, definitions, fundamental properties, and significant results from the literature related to these sequences are presented to establish a solid mathematical foundation. Structural relationships among these sequences are evaluated, and detailed discussions of their Binet formulas and various summation identities are provided. The primary focus of the study is the Gibonacci sequence, which is generalized beyond its classical definition under specific initial conditions. Analytical investigations of this generalized structure are conducted. Additionally, alternative computational methods for these sequences are proposed and supported with numerical examples. All theoretical findings are validated and illustrated through the Maple™ 17 computer algebra system. Finally, an alternative expression of a new number sequence using summation notation is explored and evaluated as a general approach.