On the characterization of periodic generalized Horadam sequences

The Horadam sequence is a direct generalization of the Fibonacci numbers in the complex plane, which depends on a family of four complex parameters: two recurrence coefficients and two initial conditions. In this article a computational matrix-based method is developed to formulate necessary and sufficient conditions for the periodicity of generalized complex Horadam sequences, which are generated by higher order recurrences for arbitrary initial conditions. The asymptotic behaviour of generalized Horadam sequences generated by roots of unity is also examined, along with upper boundaries for the disc containing periodic orbits. Some applications are suggested, along with a number of future research directions.