OFFSET
0,1
COMMENTS
This constant is transcendental (Mahler, 1929). - Amiram Eldar, Nov 14 2020
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.
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)
EXAMPLE
0.412454033640107597783361368258455283089...
In hexadecimal, .6996966996696996... .
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
EXTENSIONS
Corrected and extended by R. J. Mathar, Mar 03 2008
STATUS
approved