The claimed congruences and automaticity results for the Thue-Morse and period-doubling continued fractions are unsupported because the key induction proposition is false.
Baum and Melvin M
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On the automaticity of sequences defined by continued fractions
The claimed congruences and automaticity results for the Thue-Morse and period-doubling continued fractions are unsupported because the key induction proposition is false.