Complex Systems

Iterations, Wolfram Sequences and Approximate Closed Formulas Download PDF

Mojtaba Moniri
Department of Mathematics, Western Illinois University
Macomb, IL 61455, USA
m-moniri@wiu.edu

Abstract

Examples of computationally simplifying some sequences of non-negative integers are presented. The reduction might be at the cost of leaving out a set of exceptional inputs of zero or rather small density.

Iterations of with specific initial values x∈[-2,2] are considered. Modulo base-4 normality of , when x=0 and m is outside a set of density about , equals ; plus 1 on the exceptional set. Adding the second term of a series for , it is asked whether any exceptions remain.

Next, Wolfram sequences c, of iterated starting at 2, s of their base-2 lengths and are discussed. Under some conditions, including c not achieving a power of 2 greater than 4, with γ≈0.0972... expressible via an Odlyzko–Wilf constant. Unconditionally, γ can be removed if outside a set of density between 0.9027 and 0.9028, so is -1.