The Thue-Morse constant, also called the parity constant, is given by the concatenated digits of the Thue-Morse sequence
(1)
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(OEIS A010060) interpreted as a binary number. In, decimal, it can be written as
(2)
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(3)
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(OEIS A014571), where is the parity of (i.e., the numbers of 1s in the binary representation of , computed modulo 2).
Dekking (1977) proved that the Thue-Morse constant is transcendental, and Allouche and Shallit give a complete proof correcting a minor error of Dekking.
The Thue-Morse constant can be written in base 2 by stages by taking the previous iteration , taking the complement obtained by reversing the digits of , and appending, producing
(4)
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(5)
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(6)
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(7)
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(8)
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This can be written symbolically as
(9)
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with . Here, the complement is the number such that , which can be found from
(10)
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(11)
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(12)
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Therefore,
(13)
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and
(14)
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(15)
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The first few iterations give 0, 1/4, 3/8, 105/256, 13515/32768, ... (OEIS A074072 and A074073).
The regular continued fraction for the Thue-Morse constant is [0 2 2 2 1 4 3 5 2 1 4 2 1 5 44 1 4 1 2 4 1 1 1 5 14 1 50 15 5 1 1 1 4 2 1 4 1 43 1 4 1 2 1 3 16 1 2 1 2 1 50 1 2 424 1 2 5 2 1 1 1 5 5 2 22 5 1 1 1 1274 3 5 2 1 1 1 4 1 1 15 154 7 2 1 2 2 1 2 1 1 50 1 4 1 2 867374 1 1 1 5 5 1 1 6 1 2 7 2 1650 23 3 1 1 1 2 5 3 84 1 1 1 1284 ...] (OEIS A014572), and seems to continue with sporadic large terms in suspicious-looking patterns. A nonregular continued fraction is
(16)
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A related infinite product is
(17)
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(18)
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(19)
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(Finch 2003, p. 437).