  Let $F(X) = \sum_{n \geq 0} (-1)^{\varepsilon_n} X^{-\lambda_n}$ be a real
lacunary formal power series, where $\varepsilon_n = 0, 1$ and
$\lambda_{n+1}/\lambda_n > 2$. It is known that the denominators $Q_n(X)$ of
the convergents of its continued fraction expansion are polynomials with
coefficients $0, \pm 1$, and that the number of nonzero terms in $Q_n(X)$ is
the $n$th term of the Stern-Brocot sequence. We show that replacing the index
$n$ by any 2-adic integer $\omega$ makes sense. We prove that $Q_{\omega}(X)$
is a polynomial if and only if $\omega \in {\mathbb Z}$. In all the other cases
$Q_{\omega}(X)$ is an infinite formal power series, the algebraic properties of
which we discuss in the special case $\lambda_n = 2^{n+1} - 1$.

Number Theory;Combinatorics