On Centered Square Number

This study focuses on Centered Square Numbers (CS), a type of centered figurate number defined by a central dot surrounded by successive square layers. The research explores their properties through methods such as Ceiling Functions, Recursive Relations, Partial Sums, and Generating Functions. It also investigates their connections to other mathematical sequences and demonstrates practical applications, like solving integer equations and identifying integer side lengths in isosceles triangles. Using descriptive and expository approaches, the study clarifics patterns and mathematical reasoning behind CS.. The findings enhance understanding of their role in mathematical theory and problem-solving, while recommendations suggest extending the study into real-world applications, dynamic modeling, and exploration in non-Euclidean and quantum contexts.