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Thue-Morse Constant

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

P=0.0110100110010110100101100..._2
(1)

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

P=1/2sum_(n=0)^(infty)P(n)2^(-n)
(2)
=0.4124540336401075977...
(3)

(OEIS A014571), where P(n) is the parity of n (i.e., the numbers of 1s in the binary representation of n, 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 a_n, taking the complement a^__n obtained by reversing the digits of a_n, and appending, producing

a_0=0.0_2
(4)
a_1=0.01_2
(5)
a_2=0.0110_2
(6)
a_3=0.01101001_2
(7)
a_4=0.0110100110010110_2.
(8)

This can be written symbolically as

a_(n+1)=a_n+a^__n·2^(-2^n)
(9)

with a_0=0. Here, the complement is the number a^__n such that a_n+a^__n=0.11...1_()_(2^n)_2, which can be found from

a_n+a^__n=sum_(k=1)^(2^n)(1/2)^k
(10)
=(1-(1/2)^(2^n))/(1-1/2)-1
(11)
=1-2^(-2^n).
(12)

Therefore,

a^__n=1-2^(-2^n)-a_n,
(13)

and

a_(n+1)=a_n+(1-2^(-2^n)-a_n)2^(-2^n)
(14)
=2^(-2^(n+1))(2^(2^n)-1)(1+2^(2^n)a_n).
(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

P=1/(3-1/(2-1/(4-3/(16-(15)/(256-(255)/(65536-...)))))).
(16)

A related infinite product is

P=1/4[2-product_(n=0)^(infty)(2^(2^n)-1)/(2^(2^n))]
(17)
=1/4[2-product_(n=0)^(infty)2^(-2^n)(2^(2^n)-1)]
(18)
=1/4(2-(1·3·15·255·65535...)/(2·4·16·256·65536...))
(19)

(Finch 2003, p. 437).

See also

Digit Count, Komornik-Loreti Constant, Parity, Rabbit Constant, Thue Constant

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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." http://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. http://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."

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Thue-Morse Constant

Cite this as:

Weisstein, Eric W. "Thue-Morse Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Thue-MorseConstant.html

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