Complex Systems

Iterations, Wolfram Sequences and Approximate Closed Formulas Download PDF

Mojtaba Moniri
Department of Mathematics, Western Illinois University
Macomb, IL 61455, USA


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.