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 -