## Thue-Morse Constant

The constant also called the Parity Constant and defined by

 (1)

(Sloane's A014571), where is the Parity of . 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 , and appending, producing

 (2)

This can be written symbolically as
 (3)

with . Here, the complement is the number such that , which can be found from
 (4)

Therefore,
 (5)

and
 (6)

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 ...] (Sloane's A014572), and seems to continue with sporadic large terms in suspicious-looking patterns. A nonregular Continued Fraction is
 (7)

A related infinite product is
 (8)

The Sequence (Sloane's A010060) is known as the Thue-Morse Sequence.

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. In preparation.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Item 122, Feb. 1972.

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

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html

Sloane, N. J. A. Sequences A010060, A014571, and A014572 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.