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

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. Sm...

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Bibliographic Details
Published in:Mathematics (Basel) 2023-03, Vol.11 (5), p.1244
Main Authors: Bagdasar, Ovidiu, Chen, Minsi, Drăgan, Vasile, Ivanov, Ivan Ganchev, Popa, Ioan-Lucian
Format: Article
Language:English
Subjects:
Algorithms
complex recurrent sequences
dense orbits
Food science
Generators
geometric patterns
Horadam sequence
Investigations
Mathematical analysis
Numbers, Random
Orbits
Parameters
Periodic variations
Pseudorandom
Random numbers
Sequences
Sequences (Mathematics)
Simulation
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Summary: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 π, 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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math11051244