## Iterations, Wolfram Sequences and Approximate Closed Formulas

**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.