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.
