On the Systematic Generation of Infinite Families of Recurrence Sequences from a Closed-Form Template

This work presents a systematic investigation into the generative potential of the standard closed-form solution for linear homogeneous recurrence relations. By treating $b_n = A r_1^n + B r_2^n$ as a universal template rather than a mere solution formula, the paper formalizes the construction of infinite families of sequences. It rigorously proves that fixing characteristic roots and varying coefficients generates an infinite set of sequences sharing the same recurrence, while varying the roots themselves produces an uncountable collection of distinct recurrence families. The study further examines algebraic and asymptotic properties, integrality conditions, and the closure of these sequences under linear combinations, providing a framework for exploring the combinatorial, algebraic, and analytical landscape of recurrence sequences. This perspective highlights both the structural richness and the creative potential inherent in discrete sequence generation.