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

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.

Critical Analysis of Beta Random Variable Generation Methods

“…is satisfied, then the value X = W β+W will be accepted. Regarding the choice of θ, in [49], it was concluded that taking θ = 1/(α + β) is the best option to maximise the conditional probability of satisfying the test in Equation (9) given that the algorithm is accepted. This value for θ causes a drawback: it must be ensured that α < β.…”

“…It is at this point that we will take a closer look at some of the most important concepts related to random and pseudo-random sequences and the tests that must be carried out to check the quality of the sequences. In general, it is possible to speak of true random numbers and pseudo-random numbers (see, for example, [6][7][8][9][10][11] among others). The former are based on physical sources and do not need an initial input to be generated.…”