This paper presents a generalization of Jacobsthal and Jacobsthal Lucas numbers, introducing a broader class known as Jacobsthal and Jacobsthal Lucas \(p\) -numbers. Each instance within this class possesses a unique Binet formula, extending the classical properties of Jacobsthal numbers and Jacobsthal-Lucas to a more flexible and comprehensive framework. By deriving individual Binet formulas for each \(p\) -number, this work lays the foundation for new analytical methods that connect integer sequences, irrational proportions, and complex numbers in novel ways. These generalized formulas aim to deepen our understanding of numerical structures and open new paths for applications across coding theory, mathematical modeling, and other fields where such recursive relationships prove essential. Also, this work provides the first known closed-form Binet formulas for Jacobsthal and Jacobsthal-Lucas p-numbers, offering a novel generalization that enriches the theory of recursive integer sequences.
