TY - JOUR

T1 - On Horadam Sequences with Dense Orbits and Pseudo-Random Number Generators

AU - Bagdasar, Ovidiu

AU - Chen, Minsi

AU - Drăgan, Vasile

AU - Ivanov, Ivan Ganchev

AU - Popa, Ioan Lucian

N1 - Publisher Copyright:
© 2023 by the authors.

PY - 2023/3/4

Y1 - 2023/3/4

N2 - 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.

AB - 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.

KW - complex recurrent sequences

KW - dense orbits

KW - geometric patterns

KW - Horadam sequence

KW - random numbers

UR - http://www.scopus.com/inward/record.url?scp=85149810788&partnerID=8YFLogxK

U2 - 10.3390/math11051244

DO - 10.3390/math11051244

M3 - Article

AN - SCOPUS:85149810788

VL - 11

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 5

M1 - 1244

ER -