A321071 Twice the Thue-Morse constant (A014571).

8, 2, 4, 9, 0, 8, 0, 6, 7, 2, 8, 0, 2, 1, 5, 1, 9, 5, 5, 6, 6, 7, 2, 2, 7, 3, 6, 5, 1, 6, 9, 1, 0, 5, 6, 6, 1, 7, 8, 9, 5, 6, 7, 4, 8, 9, 1, 1, 5, 3, 9, 1, 1, 5, 1, 4, 6, 7, 5, 8, 8, 3, 0, 6, 9, 7, 5, 8, 7, 1, 8, 4, 7, 3, 1, 5, 6, 5, 1, 7, 7, 9, 2, 7, 6, 0, 9, 0, 8, 0

Offset: 0

Author

Jianing Song, Oct 27 2018

Keywords

• nonn
• cons

Comments

This is the value of the g.f. of A010060 at x = 1/2. This number is transcendental.

Examples

			Equals 2^(-1) + 2^(-2) + 2^(-4) + 2^(-7) + ... = 0.11010011001011010..._2 = 0.82490806728021519...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.8, p. 437.

Links

• Eric Weisstein's World of Mathematics, Thue-Morse Constant.
• Index entries for transcendental numbers.

Crossrefs

Cf. A000069, A001969, A010060, A014571.

Programs

Mathematica

P = FromDigits[{ThueMorse /@ Range[0, 400], 0}, 2];
RealDigits[2P, 10, 105][[1]] (* Jean-François Alcover, Apr 21 2021 *)

PARI

1-prodinf(n=0, 1-1.>>2^n)/2

Formula

Equals Sum_{k>=0} A010060(k)*2^(-k) = 1 - (Product_{k>=0} (1 - 2^(-2^k)))/2.

Equals Sum_{k>=1} 2^(-A000069(k)).