cp's OEIS Frontend

A014572 Continued fraction for Thue-Morse constant.

%I A014572 #46 Feb 16 2025 08:32:33
%S A014572 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,
%T A014572 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,
%U A014572 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
%N A014572 Continued fraction for Thue-Morse constant.
%H A014572 Harry J. Smith, <a href="/A014572/b014572.txt">Table of n, a(n) for n = 0..20000</a>
%H A014572 M. Beeler, R. W. Gosper, and R. Schroeppel, <a href="https://dspace.mit.edu/handle/1721.1/6086">HAKMEM</a>. Cambridge, MA: Mass. Institute of Technology Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972, Item 122, page 56.  Also <a href="http://www.inwap.com/pdp10/hbaker/hakmem/series.html#item122">HTML transcription</a>.
%H A014572 Y. Bugeaud, <a href="https://doi.org/10.5802/aif.2666">On the rational approximation to the Thue-Morse-Mahler numbers</a>, Ann. Inst. Fourier, Grenoble, 61 (2011), pp. 2065-2076. See p. 2068.
%H A014572 Y. Bugeaud and M. Queffélec, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Bugeaud/bugeaud3.html">On Rational Approximation of the Binary Thue-Morse-Mahler Number</a>, Journal of Integer Sequences, 16 (2013), #13.2.3.
%H A014572 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Thue-MorseConstant.html">Thue-Morse Constant.</a>
%H A014572 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A014572 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>
%e A014572 A014571 = 0.4124540336401075977833613... = 0 + 1/(2 + 1/(2 + 1/(2 + 1/(1 + ...)))). - _Harry J. Smith_, Apr 25 2009
%t A014572 P = Sum[If[OddQ[Count[IntegerDigits[n, 2], 1]], 2^(-n-1), 0], {n, 0, 400}]; ContinuedFraction[P, 91] (* _Jean-François Alcover_, Oct 23 2012 *)
%t A014572 (* ThueMorse function needs $Version >= 10.2 *)
%t A014572 P = FromDigits[{ThueMorse /@ Range[0, 400], 0}, 2];
%t A014572 ContinuedFraction[P, 91] (* _Jean-François Alcover_, Jan 30 2020 *)
%o A014572 (PARI) { allocatemem(932245000); default(realprecision, 21000); x=0.0; m=70000; for (n=1, m-1, x=x+x; x=x+sum(k=0, length(binary(n))-1, bittest(n, k))%2); x=x/2^m; y=contfrac(x); for (n=1, 20001, write("b014572.txt", n-1, " ", y[n])); } \\ _Harry J. Smith_, Apr 25 2009
%Y A014572 Cf. A014571.
%K A014572 nonn,cofr
%O A014572 0,2
%A A014572 _Eric W. Weisstein_