The 3n+1 Problem & Beatty Sequences

[Joe Horn]

(08-26-2014 09:45 PM)Jim Horn Wrote:  No, nor do I understand your observation completely from the above description. But if that will be touched on in your HHC2014 presentation, I'll be taking careful notes! Great find...

Yes, it'll be the crux of my Hailstone Numbers talk at HHC 2014. Sorry for not explaining it more clearly here, but this margin is too small to contain the whole thing.

Sneak preview: Make a list of n*log(2)/log(3) where n increments by 1 from 1 to something big. Now replace all those numbers with their ceiling. Now subtract each from the number before it (the first finite difference; aka DeltaLIST in RPL). You'll get a list that starts like this:

{ 1 1 0 1 1 0 1 1 0 1 0 ... }

Exactly the same non-repeating pattern (no matter how far out you go) appears as the "parity sequence" (aka parity vector) obtained by iterating T(n) which is defined as: \begin{equation}T(n) =\begin{cases}\frac{3n+1}{2} & n\equiv 1 \, \big(mod \, 2\big)\\\frac{n}{2} & n \equiv 0 \, \big(mod \, 2\big)\end{cases}\end{equation}

But wait! There's more! Lots more. At HHC 2014.