We construct a random model to study the distribution of class numbers inspecial families of real quadratic fields $\mathbb Q(\sqrt d)$ arising fromcontinued fractions. These families are obtained by considering periodiccontinued fraction expansions of the form $\sqrt {D(n)}=[f(n), [u_1, u_2,\dots, u_{s-1}, 2f(n)]]$ with fixed coefficients $u_1, \dots, u_{s-1}$ andgeneralize well-known families such as Chowla's $4n^2+1$, for which analogousresults were recently proved by Dahl and Lamzouri.
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