Is there anything known about the asymptotic expected growth of $card\{a\_0,a\_1,...,a\_n\}$, where the $a\_i$ are the first $n$ coefficient of a continued fraction $[a\_0; a\_1, ... ]$?



Expected Cardinality of the First n Coefficients of a Continued Fraction
Since the question asks about an expectation, one needs to specify a probability space of continued fractions. It seems the most natural such space is the interval $[0,1)$ with the density function $1/(1+x)$ that is the natural invariant measure in the study of continued fractions. It is known (Khinchin) that the probability that $a\_i=r$ is equal to $\log(1+\frac1{r(r+2)})/\log 2$. Therefore the expected number of distinct $a\_i$ among $a\_1,\dots,a\_k$ is (by linearity of expectation)
$$
\sum\_{r=1}^\infty \bigg( 1 - \bigg(1- \frac{\log(1+\frac1{r(r+2)})}{\log 2} \bigg)^k \bigg).
$$
(The $r$th summand is the probability that $r$ occurs at least once among $a\_1,\dots,a\_k$.) When $\frac1{r(r+2)}$ is small compared with $\frac1k$, then the $r$th summand is approximately $k\log(1+\frac1{r(r+2)})/\log 2 \asymp \frac k{r^2}$; the contribution of such terms where $r>R$ say (with $R$ large compared with $\sqrt k$) is $O(k/R)$. The contribution of the initial $\sqrt k$ or so terms should have order of magnitude $\sqrt k$. So I suspect that **the expected number of distinct $a\_i$ among the first $k$ partial quotients is asymptotic to some constant times $\sqrt k$**. Numerical experimentation is reasonably consistent with this, with the constant quite close to $2.1$.


