Thue-Morse Constant

The Thue-Morse constant, also called the parity constant, is given by the concatenated digits of the Thue-Morse sequence

(1)

(OEIS A010060) interpreted as a binary number. In decimal notation, it can be written as

			(2)
			(3)

(OEIS A014571), where is the binary parity of a nonnegative integer (i.e., the number of 1s in the binary representation of , computed modulo 2).

Dekking (1977) proved that the Thue-Morse constant is transcendental. Allouche and Shallit (2003) give a corrected version of Dekking's proof.

The Thue-Morse constant can be written in binary (base 2) by stages by taking the previous iteration , taking the ones' complement obtained by exchanging 0 and 1 in the -digit fractional word of , and appending, producing

			(4)
			(5)
			(6)
			(7)
			(8)

This can be written symbolically as

(9)

for integers , with . Here, the ones' complement is the number such that , which can be found from

			(10)
			(11)
			(12)

Therefore,

(13)

and

			(14)
			(15)

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)

A related infinite product is

			(17)
			(18)
			(19)

(Finch 2003, p. 437).