Iterations, Wolfram Sequences and Approximate Closed Formulas

Mojtaba Moniri
2017 Complex Systems  
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 a m  m∈ℕ of 2 + x with specific initial values x ∈ -2, 2 are considered. Modulo base-4 normality of 1 π 2 , when x  0 and m is outside a set of density about π 2 ; plus 1 on the exceptional set. Adding the second term of a series for 1 2-a m  1 4 csc 2  π 2 m+2  , it is
more » ... 2 m+2  , it is asked whether any exceptions remain. Next, Wolfram sequences c, of iterated  3 2 x starting at 2, s of their base-2 lengths and r m  min k s k ≥ m are discussed. Under some conditions, including c not achieving a power of 2 greater than 4, r m  m log 2 3 2 + γ -1 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.
doi:10.25088/complexsystems.26.2.97 fatcat:x4lvxcwejbhy5isz63a26ztgoy