On some properties and applications of Horadam sequences

The Horadam sequence is a generalization of the Fibonacci numbers in the complex plane, depending on a family of four complex parameters: two recurrence coefficients and two initial conditions. The necessary and sufficient periodicity conditions formulated in [1] are used to enumerate all Horadam sequences with a given period [2]. The geometry of periodic orbits is analyzed, where regular star-polygons, bi-partite digraphs and multi-symmetric patterns are recovered. A number of periodic and nonperiodic Horadam patterns are presented (convergent, divergent or dense), along with a Horadam-based pseudo-random generator [3]. Periodicity conditions for generalised Horadam sequences (produced by higher order recurrences) are also formulated [4]. References. [1] O. Bagdasar and P. J. Larcombe, On the characterization of periodic complex Horadam sequences, Fibonacci Quart 51 (1) (2013) 28-37. [2] O. Bagdasar and P. J. Larcombe, On the number of complex Horadam sequences with a fixed period, Fibonacci Quart 51 (4) (2013) 339-347. [3] O. Bagdasar and M. Chen, A Horadam-based Pseudo-Random Number generator, Proceedings of 16th UKSim Cambridge, (2014) 226-230. [4] O. Bagdasar and P. J. Larcombe, On the characterization of periodic generalized complex Horadam sequences, J Differ Equ Appl, 20, (2014) 1069-1090.