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, it can be written as

			(2)
			(3)

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

This can be written symbolically as

(9)

with . Here, the 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).

REFERENCES:

Allouche, J. P.; Arnold, A.; Berstel, J.; Brlek, S.; Jockusch, W.; Plouffe, S.; and Sagan, B. "A Relative of the Thue-Morse Sequence." Discr. Math. 139, 455-461, 1995.

Allouche, J. P. and Shallit, J. "The Ubiquitous Prouhet-Thue-Morse Sequence." https://www.math.uwaterloo.ca/~shallit/Papers/ubiq.ps.

Dekking, F. M. "Transcendence du nombre de Thue-Morse." Comptes Rendus de l'Academie des Sciences de Paris 285, 157-160, 1977.

Finch, S. R. "Prouhet-Thue-Morse Constant." §6.8 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 436-441, 2003.

Goldstein, S.; Kelly, K. A.; and Speer, E. R. "The Fractal Structure of Rarefied Sums of the Thue-Morse Sequence." J. Number Th. 42, 1-19, 1992.

Schroeppel, R. and Gosper, R. W. Item 122 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 56-57, Feb. 1972. https://www.inwap.com/pdp10/hbaker/hakmem/series.html#item122.

Sloane, N. J. A. Sequences A010060, A014571, A014572, A074072, and A074073 in "The On-Line Encyclopedia of Integer Sequences."