Closed-Form Solutions of Second-Order Leonardo-Type Sequences: Homogeneous Counterparts in Jacobsthal and Mersenne Numbers
Yuksel Soykan *
Department of Mathematics, Faculty of ScienceZonguldak Bulent Ecevit University, 67100, Zonguldak, Turkey.
*Author to whom correspondence should be addressed.
Abstract
The objective of this study is to derive explicit closed-form solutions for second-order nonhomogeneous linear recurrence relations with polynomial inputs, formulated as generalized Leonardo-type sequences. A central aspect of the framework is the multiplicity parameter r, which measures the occurrence of the root 1 in the characteristic equation. This parameter determines whether the recurrence falls into the non-resonant case (r = 0, no root equal to 1) or the resonant case (r = 1, unity as a simple root), with corresponding adjustments in the construction of particular solutions.
Within this setting, we highlight two principal families: the generalized Jacobsthal sequences, where the characteristic roots are {2, −1} and thus r = 0, and the generalized Mersenne sequences, where the roots are {2, 1} and hence r = 1. In both cases, closed-form solutions are obtained under polynomial inputs of degrees s = 0, 1, 2, 3, 4, 5, 6, 7, covering constant through septic forcing terms. These results clarify how root multiplicity and polynomial degree jointly shape the explicit formulas, while the homogeneous counterparts (Jacobsthal, Jacobsthal-Lucas, Mersenne, and Mersenne-Lucas sequences) emerge naturally when the input polynomial is suppressed.
The study thus provides a unified framework that connects classical integer sequences with their nonhomogeneous extensions, offering resonance-aware closed forms that are both theoretically significant and pedagogically accessible.
Keywords: Jacobsthal numbers, Mersenne numbers, Leonardo numbers, nonhomogeneous recurrence relations, recurrence relations, closed-form solutions, particular solutions