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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 $\\omega \\in {\\mathbb Z}$. 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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 $\\omega \\in {\\mathbb Z}$. 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