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

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  • Ovidiu Bagdasar

    (School of Computing and Engineering, University of Derby, Derby DE22 1GB, UK
    Department of Mathematics, Faculty of Exact Sciences, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania)

  • Minsi Chen

    (Department of Computer Science, University of Huddersfield, Huddersfield HD1 3DH, UK)

  • Vasile Drăgan

    (“Simion Stoilow” Institute of Mathematics, Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
    Academy of the Romanian Scientists, Str. Ilfov, Nr. 3, 50044 Bucharest, Romania)

  • Ivan Ganchev Ivanov

    (Faculty of Economics and Business Administration, Sofia University “St. Kliment Ohridski”, 125 Tzarigradsko Chaussee Blvd., Bl. 3, 1113 Sofia, Bulgaria)

  • Ioan-Lucian Popa

    (Department of Mathematics, Faculty of Exact Sciences, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania
    Faculty of Mathematics and Computer Science, Transilvania University of Braşov, Iuliu Maniu Street 50, 500091 Braşov, Romania)

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.

Suggested Citation

  • Ovidiu Bagdasar & Minsi Chen & Vasile Drăgan & Ivan Ganchev Ivanov & Ioan-Lucian Popa, 2023. "On Horadam Sequences with Dense Orbits and Pseudo-Random Number Generators," Mathematics, MDPI, vol. 11(5), pages 1-16, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:5:p:1244-:d:1087646
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    References listed on IDEAS

    as
    1. Hellekalek, P., 1998. "Good random number generators are (not so) easy to find," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 46(5), pages 485-505.
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    Cited by:

    1. Elena Almaraz Luengo & Carlos Gragera, 2023. "Critical Analysis of Beta Random Variable Generation Methods," Mathematics, MDPI, vol. 11(24), pages 1-31, December.

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