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


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?


This was first done in my paper, Simple Continued Fractions for Some Irrational Numbers II, published in J. Number Theory 14 (1982), 228-231.

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

  • 2
    $\begingroup$ It's a shame your paper is not the references of the webpage! It is also absent from the references in oeis.org/A058304 ... Many thanks! $\endgroup$ – ARG Apr 30 '17 at 10:03

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