Translation of Ruggles' Identity F(n+k) = L(k)F(n) + (-1)^(k+1)F(n-k) into a Simplified Applied Method for Generating Fibonacci Numbers via Lucas Constants and Two Parallel Sliding Fibonacci Sequences

Abstract This paper introduces an original applied method for generating Fibonacci sequences based on Lucas constants and a "parallel sliding" technique. The author provides a practical transition from the theoretical Ruggles' Identity (1963) to a dynamic algorithmic model optimized for spreadsheet environments such as Microsoft Excel. The method is built upon a "Pivot Axis" principle, where a selected Lucas number acts as a fixed constant while two Fibonacci sequences slide synchronously starting from different positions (the selected index and the first index, respectively). This approach eliminates the common issue of index asymmetry and the need for zero or negative indices found in classical formulas. By establishing a direct correlation between the Lucas index parity and the operational sign (addition or subtraction), the method offers a reliable, error-resistant, and highly visual tool for both educational purposes and computational number theory. The paper includes five detailed demonstration tables illustrating the "sliding resonance" across various Lucas constants.