%I
%S 1,2,7,106,27031,1771476586,7608434000728254871,
%T 140350834813144189858090274002849666666,
%U 47758914269546354982683078068829456704164423862093743397580034411621752859031
%N Numerators of fractions with denominators A000215(n) approximating the ThueMorse constant
%C One can prove that if in the sequence of numbers N for which A010060(N+2^n)= A010060(N) you replace the odious (evil) terms by 1's (0's), then we obtain 2^(n+1)periodic (0,1)sequence; if you write it in the form .xx...,i.e., as a binary infinite fraction, then the corresponding fraction has the form a(n)/A000215(n). These fractions very fast converge to the ThueMorse constant .4124540336401...; e.g a(5)/(2^32+1) approximates this constant up to 10^(9). These approximations differ from A074072A074073. Conjecture. For n>=1, the fraction a(n)/A000215(n) is a convergent corresponding to the continued fraction for the ThueMorse constant.
%H V. Shevelev, <a href="http://arXiv.org/abs/0907.0880">Equations of the form t(x+a)=t(x) and t(x+a)=1t(x) for ThueMorse sequence</a>,
%F a(1)=2, and, for n>=2, a(n) = 1 + (2^(2^(n1))1) * a(n1).
%o (PARI) a(n)=if(n<=1, [1,2][n+1], 1+(2^(2^(n1))1)*a(n1)); /* _Joerg Arndt_, Mar 11 2013 */
%Y Cf. A010060, A000215, A085394, A085395, A081706, A161627, A161639, A074072, A074073.
%K nonn,uned
%O 0,2
%A _Vladimir Shevelev_, Jul 08 2009, Jul 14 2009
%E Added more terms, _Joerg Arndt_, Mar 11 2013
