[seqfan] Re: Binary Complement Sequences

[Arthur O'Dwyer]

On Thu, Dec 29, 2022 at 1:18 PM Allan Wechsler <acwacw at gmail.com> wrote:

> It would be worthwhile comparing the trajectory plots of 819991 with those
> for 425750. For all we know, the trajectories merge around the seventeen
> millionth step, and if they do, it won't be necessary to run 819991 to
> completion. In fact, 819991 might be an early iterate of 425750.
>

and Tim Peters wrote:
> Iterating 717657 for 46 steps coincides with 820560's successor [namely,
1732623]

I agree that more exploration isn't as useful as actually trying to *prove*
universal convergence/divergence/cycle (by contradiction or by
construction).

But since people are talking about exploration...

- When we say "N reaches zero," we mean "N reaches anything less than N,"
right? I mean, if you're searching the number line in increasing order, you
know that as soon as N reaches a number you've seen before, you're done
(because all the numbers you've ever seen, converge; and you've already
seen all numbers less than N).
- This relates to Allan's suggestion that "we can catch a lot of
Methuselahs early, by keeping a table of, say, the billionth iterates of
long-lasting seeds."  That's a technique known as rainbow tables
<https://en.wikipedia.org/wiki/Rainbow_table>, and sounds like a good idea
to me.
- Tom, is your code available somewhere to look at and/or modify?

–Arthur