I've been interested in the numbers of this form because it can be proved that for integer $a \geq 2$ all of them are irrational: $$x_a=\sum_{n=0}^\infty \frac{1}{a^{2^n}}$$
They satisfy the conditions listed in this paper: The Approximation of Numbers as Sums of Reciprocals. This is related to my other question.
Now I decided to look at simple continued fractions of such numbers and noticed a surprising thing. For each $a$ I checked the CF entries consist of only three numbers (except the first entry, which is why I show $1/x$ instead of $x$):
$$x_2=\sum_{n=0}^\infty \frac{1}{2^{2^n}}=0.8164215090218931437080797375305252217$$
$$\frac{1}{x_2}=1+\cfrac{1}{4+\cfrac{1}{2+\cfrac{1}{4+\cfrac{1}{4+\cfrac{1}{6+\dots}}}}}$$
Writing the CF in the more convenient form we obtain for $200$ digits:
$1/x_2=$[1; 4, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 6, 4, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 6, 4, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 6, 4, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 4, 6, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 4, 6, 4, 2, 6, 4, 2, 4, 6, 4, 4, 2, 4,...]
Clearly, all of the entries are $2,4$ or $6$.
The same goes for other examples:
$1/x_3=$[2; 5, 3, 3, 1, 3, 5, 3, 1, 5, 3, 1, 3, 3, 5, 3, 1, 5, 3, 3, 1, 3, 5, 1, 3, 5, 3, 1, 3, 3, 5, 3, 1, 5, 3, 3, 1, 3, 5, 3, 1, 5, 3, 1, 3, 3, 5, 1, 3, 5, 3, 3, 1, 3, 5, 1, 3, 5, 3, 1, 3, 3, 5, 3, 1, 5, 3, 3, 1, 3, 5, 3, 1, 5, 3, 1, 3, 3, 5, 3, 1, 5, 3, 3, 1, 3, 5, 1, 3, 5, 3, 1, 3, 3, 5, 1, 3, 5, 3, 3, 1, 3, 5, 3, 1, 5, 3, 1, 3, 3, 5, 1, 3, 5, 3, 3, 1, 3, 5, 1, 3, 5, 3, 1, 3, 3, 5, 3, 1, 5, 3, 3, 1, 3, 5, 3, 1, 5, 3, 1, 3, 3, 5, 3, 1, 5, 3, 3, 1, 3, 5, 1, 3, 5, 3, 1, 3, 3, 5, ,...]
$1/x_5=$[4; 7, 5, 5, 3, 5, 7, 5, 3, 7, 5, 3, 5, 5, 7, 5, 3, 7, 5, 5, 3, 5, 7, 3, 5, 7, 5, 3, 5, 5, 7, 5, 3, 7, 5, 5, 3, 5, 7, 5, 3, 7, 5, 3, 5, 5, 7, 3, 5, 7, 5, 5, 3, 5, 7, 3, 5, 7, 5, 3, 5, 5, 7, 5, 3, 7, 5, 5, 3, 5, 7, 5, 3, 7, 5, 3, 5, 5, 7, 5, 3, 7, 5, 5, 3, 5, 7, 3, 5, 7, 5, 3, 5, 5, 7, 3, 5, 7, 5, 5, 3, 5, 7, 5, 3, 7, 5, 3, 5, 5, 7, 3, 5, 7, 5, 5, 3, 5, 7, 3, 5, 7, 5, 3, 5, 5, 7, 5, 3, 7, 5, 5, 3, 5, 7, 5, 3, 7, 5,...]
How can this phenomenon be explained?
The further questions are:
Can we prove that all the CF entries for these numbers belong to a fixed set of three integers?
If so, can we make any conclusions about transcendentality of these numbers?
The implications are interesting. It is conjectured that algebraic numbers of degree $>2$ should have arbitrarily large CF entries at some point. See this paper. Meanwhile we know, that for degree $2$ the CF is (eventually) periodic.
Notice also the same 'pattern' which goes for the examples with odd $a$ here. We have a list of CF entries going the same way.
If we subtract the list of entries for $a=3$ from the list of entries for $a=5$, we obtain:
$$L_5-L_3=[0;2,2,2,2,2,2,2,2,2,...]$$
$$L_7-L_3=[0;4,4,4,4,4,4,4,4,4,...]$$
$$L_{113}-L_3=[0;110,110,110,110,110,110,...]$$
For $6$ and $2$ it goes the same way, but not for $4$ and $2$, there is some 'scrambling' there.
How can we prove/explain this facts? The same pattern for different $a$ seems very strange to me, especially if the numbers are transcendental.
Basically, if this turns out to be true, then from the CF for $x_3$ we will immediately obtain all the CFs for every $x_{2n+1}$
Important update. See http://oeis.org/A004200 for the case $a=3$, it seems like these continued fractions have pattern. And the following paper is linked: Simple continued fractions for some irrational numbers
The pattern is the same for every $a$ except $2$, so not only for the odd $a$.
Morevoer, look at the continued fractions for:
$$y_{ap}=a^p x_a$$
For integer $p$ you will notice a very apparent pattern.
My questions are largely answered by the linked paper. I will try to write up a short summary and post it as an answer, but anyone is three to do it before me.
Turns out a very close question was asked before Continued fraction for $c= \sum_{k=0}^\infty \frac 1{2^{2^k}} $ - is there a systematic expression?