Particular Solutions of Generalized Leonardo-Type Recurrences with Exponential Input

Recurrence sequences with exponential input terms arise naturally in symbolic recurrence theory and extend classical families such as Fibonacci and Lucas. In earlier work, we established a general theorem that provides explicit iterative formulas for computing particular solutions of generalized Leonardo-type recurrences. The present article focuses on concrete applications of that result. We derive and present explicit particular solutions for the special cases of orders m = 1, 2, 3, 4, 5, 6 illustrating how the general framework specializes to low-order recurrences. These examples highlight the role of characteristic polynomials, root multiplicities, and resonance phenomena, while offering closed-form constructions that bridge classical recurrence identities with modern symbolic methods.