On Horadam Sequences with Dense Orbits and Pseudo-Random Number Generators

Ovidiu Bagdasar, Minsi Chen, Vasile Drăgan, Ivan Ganchev Ivanov, Ioan Lucian Popa

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


Horadam sequence is a general recurrence of second order in the complex plane, depending on four complex parameters (two initial values and two recurrence coefficients). These sequences have been investigated over more than 60 years, but new properties and applications are still being discovered. Small parameter variations may dramatically impact the sequence orbits, generating numerous patterns: periodic, convergent, divergent, or dense within one dimensional curves. Here we explore Horadam sequences whose orbit is dense within a 2D region of the complex plane, while the complex argument is uniformly distributed in an interval. This enables the design of a pseudo-random number generator (PRNG) for the uniform distribution, for which we test periodicity, correlation, Monte Carlo estimation of (Formula presented.), and the NIST battery of tests. We then calculate the probability distribution of the radii of the sequence terms of Horadam sequences. Finally, we propose extensions of these results for generalized Horadam sequences of third order.

Original languageEnglish
Article number1244
Number of pages16
Issue number5
Publication statusPublished - 4 Mar 2023

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