Allan's reply caused me to remember that Wolfram's "A New Kind of Science" has a brief discussion of things like this. Maybe I can even find my copy at home. In fact, I also kind of recall having a conversation with Cris Moore and Stephen Wolfram in which Cris got Stephen to concede that some of the stuff related to not-quite-chaotic carrying is a counterexample to Wolfram's typology. (Wolfram's reaction was that of a physicist, not a mathematician; he felt with a principle that had only one or two exceptions was a pretty good principle.) Cris, do you remember more? Jim On Wednesday, July 13, 2016, Allan Wechsler <acwacw@gmail.com> wrote:
The traffic jam of carries in the middle isn't too surprising -- I would be amazed if you could ever get anything sensible out of them. Similarly, the growing fixed zones at opposite ends of the number are not totally unexpected: the one on the left is due to the fact that Prod(1-10^(-k)) converges, while the one on the right is a similar phenomenon happening in the ring of 10-adic numbers.
What made my jaw drop was the intermediate zone, between that growing fixed outer crust and the totally chaotic core. There is a "mantle" of stuff that clearly exhibits patterns, but the patterns shift slowly -- blocks of 0s growing by one at every step, for instance. And I don't understand why it exists at all. If we want to adopt Jim's "phase" metaphor, I understand the solid crust and the gaseous core, but not the liquid mantle.
On Wed, Jul 13, 2016 at 11:15 AM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
Thanks, Neil! I don't know why I didn't think to check the OEIS.
Jim, that is https://oeis.org/A027828.
Actually it's https://oeis.org/A027878.
If you find out something interesting, please add a comment there. There is a certain amount of info there already, of course.
I doesn't appear that anyone has published anything about the middle digits. When I get a chance I might tally the hundred middle digits for progressively larger n, to see if some sort of asymptotic equidistribution appears to be going on there.
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