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> 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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