This article considers a generalized Fibonacci sequence $\{V_{n}\}$ with general initial conditions, $V_{0}=a$, $V_{1}=b$, and a versatile recurrence relation $V_{n}=pV_{n-1}+qV_{n-2}$, where $n\ge 2$ and $a, b, p$ and $q$ are any non-zero real numbers. The generating function and Binet formula for this generalized sequence are derived. This generalization encompasses various well-known sequences, including their generating functions and Binet formulas as special cases. Numerous new properties of these sequences are studied and investigated.