On Horadam Sequences with Dense Orbits and Pseudo-Random Number Generators
Journal article
Bagdasar, O., Chen, M, Drăgan, V., Ivanov, I.G. and Popa, I.-L. 2023. On Horadam Sequences with Dense Orbits and Pseudo-Random Number Generators. Mathematics. 11 (5), pp. 1-16. https://doi.org/10.3390/math11051244
| Authors | Bagdasar, O., Chen, M, Drăgan, V., Ivanov, I.G. and Popa, I.-L. |
|---|---|
| Abstract | 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. |
| Keywords | random numbers; geometric patterns; complex recurrent sequences; Horadam sequence; dense orbits |
| Year | 2023 |
| Journal | Mathematics |
| Journal citation | 11 (5), pp. 1-16 |
| Publisher | MDPI |
| ISSN | 2227-7390 |
| Digital Object Identifier (DOI) | https://doi.org/10.3390/math11051244 |
| Web address (URL) | https://www.mdpi.com/2227-7390/11/5/1244 |
| Accepted author manuscript | File Access Level Open |
| Publisher's version | License File Access Level Open |
| Output status | Published |
| Publication dates | |
| Online | 04 Mar 2023 |
| Publication process dates | |
| Accepted | 02 Mar 2023 |
| Deposited | 26 Apr 2023 |
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Bagdasar, Ovidiu and Larcombe, Peter J. 2013. On the characterization of periodic complex Horadam sequences. Fibonacci Quarterly.On a result of Bunder involving Horadam sequences: A proof and generalization
Larcombe, Peter J. and Bagdasar, Ovidiu 2013. On a result of Bunder involving Horadam sequences: A proof and generalization. Fibonacci Quarterly.Influence discovery in semantic networks: An initial approach
Trovati, M. and Bagdasar, O. 2014. Influence discovery in semantic networks: An initial approach. IEEE. https://doi.org/10.1109/uksim.2014.48Complementary iInequalities involving the Stolarsky Mean
Bagdasar, Ovidiu 2010. Complementary iInequalities involving the Stolarsky Mean. Journal of Inequalities and Applications. https://doi.org/10.1155/2010/492570Concise computer mathematics: tutorials on theory and problems
Bagdasar, Ovidiu 2013. Concise computer mathematics: tutorials on theory and problems. Springer.On the characterization of periodic generalized Horadam sequences
Bagdasar, Ovidiu and Larcombe, Peter J. 2014. On the characterization of periodic generalized Horadam sequences. Journal of Difference Equations and Applications. https://doi.org/10.1080/10236198.2014.891022Local maximum points of explicitly quasiconvex functions
Bagdasar, Ovidiu and Popovici, Nicolae 2014. Local maximum points of explicitly quasiconvex functions. Optimization Letters. https://doi.org/10.1007/s11590-014-0781-3A horadam-based pseudo-random number generator
Bagdasar, O.D., Chen, M. and Bagdasar, O. 2014. A horadam-based pseudo-random number generator. IEEE. https://doi.org/10.1109/uksim.2014.55On the structure of periodic complex Horadam orbits
Bagdasar, Ovidiu D., Larcombe, Peter J., Anjum, Ashiq and Bagdasar, O. 2016. On the structure of periodic complex Horadam orbits. Carpathian Journal of Mathematics.Characterizations of Generalized Exponential Trichotomies for Linear Discrete-time Systems
Popa, I.-L., Ceauşu, T., Bagdasar, O. and Agarwal, R.P. 2016. Characterizations of Generalized Exponential Trichotomies for Linear Discrete-time Systems. Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica. 27 (2), pp. 153-166. https://doi.org/10.2478/auom-2019-0025On some functions involving the lcm and gcd of integer tuples
Bagdasar, Ovidiu 2014. On some functions involving the lcm and gcd of integer tuples. Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and mechanics. https://doi.org/10.5937/SPSUNP1402091B