“Thue-Morse Constant”的概念、定义、习题解题方法及翻译-数学辞典

单词

Thue-Morse Constant

释义

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 isTranscendental, and Allouche and Shallit give a complete proof correcting a minorerror 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 5 141 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 15154 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 tocontinue 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.

See also Rabbit Constant, Thue Constant

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, andA014572in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.

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