A friend and I were discussing the properties of continued fractions (as "best" approximations). For fun, we checked the continued fractions of Liouville's constant. The terms in the sequence fit a very clear pattern, see


<http://mathworld.wolfram.com/LiouvillesConstant.html>


Is there an actual reference to show that the sequence is a nested sequence?
Here are more specific questions that focus on certain issues. Let $x = \sum\_{i=1}^{\infty} a^{-i!}$. Is there a reference for 


1. The only terms in the continued fraction are 
$$
1, a-2, a, a+1 \textrm{ and } a^k-1 \textrm{ for some } k \geq 1.
$$
2. The $n^\textrm{th}$ incrementally largest term (considering only those entirely of the form $a^k-1$) occurs precisely at position $2^n-1$.
3. The exponent $k$ in the largest term at position $2^n-1$ is $n!\cdot (n-1)$.
4. What is the algorithm to build the sequence as a nested sequence?



Continued fraction of Liouville's constant
This was first done in my paper, [Simple Continued Fractions for Some Irrational Numbers II](http://www.sciencedirect.com/science/article/pii/0022314X82900476), published in J. Number Theory 14 (1982), 228-231.


From the proof there you can deduce the 4 things you listed.


