A014571 Consider the Morse-Thue sequence (A010060) as defining a binary constant and convert it to decimal.

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

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• cons

Comments

This constant is transcendental (Mahler, 1929). - Amiram Eldar, Nov 14 2020

Examples

			0.412454033640107597783361368258455283089...
In hexadecimal, .6996966996696996... .
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.8 Prouhet-Thue-Morse Constant, p. 437.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• Boris Adamczewski and Yann Bugeaud, A short proof of the transcendence of Thue-Morse continued fractions, The American Mathematical Monthly, Vol. 114, No. 6 (2007), pp. 536-540; alternative link.
• Jean-Paul Allouche and Jeffrey Shallit, The ubiquitous Prouhet-Thue-Morse sequence, in: C. Ding, T. Helleseth, and H. Niederreiter (eds.), Sequences and their applications, Springer, London, 1999, pp. 1-16; alternative link.
• Joerg Arndt, Matters Computational (The Fxtbook), p.726 ff.
• Michel Dekking, Transcendance du nombre de Thue-Morse, Comptes Rendus de l'Academie des Sciences de Paris, Série A, Vol. 285 (1977) A157-A160.
• Arturas Dubickas, On the distance from a rational power to the nearest integer, Journal of Number Theory, Volume 117, Issue 1, March 2006, Pages 222-239.
• Kurt Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Mathematische Annalen, Vol. 101 (1929), pp. 342-366, alternative link.
• R. Schroeppel and R. W. Gosper, HACKMEM #122 (1972).
• Eric Weisstein's World of Mathematics, Thue-Morse Constant.
• Index entries for transcendental numbers

Crossrefs

Cf. A000069, A001969, A010060, A058631, A215016, A247950.

Programs

Maple

A014571 := proc()
    local nlim,aold,a ;
    nlim := ilog2(10^Digits) ;
    aold := add( A010060(n)/2^n,n=0..nlim) ;
    a := 0.0 ;
    while abs(a-aold) > abs(a)/10^(Digits-3) do
        aold := a;
        nlim := nlim+200 ;
        a := add( A010060(n)/2^n,n=0..nlim) ;
    od:
    evalf(%/2) ;
end:
A014571() ; # R. J. Mathar, Mar 03 2008

Mathematica

digits = 105; t[0] = 0; t[n_?EvenQ] := t[n] = t[n/2]; t[n_?OddQ] := t[n] = 1-t[(n-1)/2]; FromDigits[{t /@ Range[digits*Log[10]/Log[2] // Ceiling], -1}, 2] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 20 2014 *)
1/2-1/4*Product[1-2^(-2^k), {k, 0, Infinity}] // N[#, 105]& // RealDigits // First (* Jean-François Alcover, May 15 2014, after Steven Finch *)
(* ThueMorse function needs $Version >= 10.2 *)
P = FromDigits[{ThueMorse /@ Range[0, 400], 0}, 2];
RealDigits[P, 10, 105][[1]] (* Jean-François Alcover, Jan 30 2020 *)

PARI

default(realprecision, 20080); x=0.0; m=67000; for (n=1, m-1, x=x+x; x=x+sum(k=0, length(binary(n))-1, bittest(n, k))%2); x=10*x/2^m; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b014571.txt", n, " ", d)); \\ Harry J. Smith, Apr 25 2009

PARI

1/2-prodinf(n=0,1-1.>>2^n)/4 \\ Charles R Greathouse IV, Jul 31 2012

Formula

Equals Sum_{k>=0} A010060(n)*2^(-(k+1)). [Corrected by Jianing Song, Oct 27 2018]

Equals Sum_{k>=1} 2^(-(A000069(k)+1)). - Jianing Song, Oct 27 2018

From Amiram Eldar, Nov 14 2020: (Start)

Equals 1/2 - (1/4) * A215016.

Equals 1/(3 - 1/A247950). (End)

Extensions

Corrected and extended by R. J. Mathar, Mar 03 2008