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We characterize functions in $h^\\infty_v$ that are represented by Hadamard gap series in terms of their coefficients, and as a corollary we obtain a characterization of Hadamard gap series in Bloch-type spaces for weights with a doubling property. We show that if $u\\in h^\\infty_v$ is represented by a Hadamard gap series, then $u $ will grow slower than $v$ or oscillate along almost all radii. 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