[seqfan] Fwd: Irregular triple fractal

[M. F. Hasler]

On Wed, Mar 27, 2019 at 9:32 PM Éric Angelini  wrote:

> a beautiful (!) sequence (which will be submitted tomorrow to the OEIS)
> -- sequence with a large amount of unsolved questions (regularity, runs
> of 0s and 1s, possible loop, etc.)
>

 The sequence is https://oeis.org/A307183

FWIW, I have added very simple PARI code to compute the sequence, following
the reasoning given in a comment (not yet approved), which shows how the
constraint (underlined and non-underlined subsequence are both equal to the
original) determines the whole sequence uniquely, including the initial
terms. There cannot be any other sequence with this property.

That said, most of the cited "unsolved questions" remain unexplained, among
others, why (or whether) the asymptotic of 1's appears to converge to
something near log(2) = 0.693...
Indeed a very nice sequence - idea !

[Lars Blomberg]

> For e=(3,4,5,6,7,8,9) the 10^e terms contain
> (70,696,6928,69249,693289,6941990,69507187,694529637) ones.
> The fraction of ones seem to be rather constant, but the are subtle
> differences from any regular trend.

(...)

I cannot imagine why this should be so, and one might suspect the
> algorithm, but as I said above, it is not at all complicated and I can’t
> find anything wrong with it.
>
> [Explanation]:
> Start reading the sequence:
> ---if a(n) = 0, underline a(n+1);
> ---if a(n) = 1, underline a(n+2).
> The underlined terms will reproduce the starting sequence.
> The not underlined terms too.
>