This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A162634 #12 Jul 22 2025 06:55:57 %S A162634 1,2,7,106,27031,1771476586,7608434000728254871, %T A162634 140350834813144189858090274002849666666, %U A162634 47758914269546354982683078068829456704164423862093743397580034411621752859031 %N A162634 Numerators of fractions with denominators A000215(n) approximating the Thue-Morse constant. %C A162634 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 Thue-Morse constant .4124540336401...; e.g a(5)/(2^32+1) approximates this constant up to 10^(-9). These approximations differ from A074072-A074073. Conjecture. For n>=1, the fraction a(n)/A000215(n) is a convergent corresponding to the continued fraction for the Thue-Morse constant. %H A162634 V. Shevelev, <a href="http://arXiv.org/abs/0907.0880">Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence</a>, %F A162634 a(1)=2, and, for n>=2, a(n) = 1 + (2^(2^(n-1))-1) * a(n-1). %o A162634 (PARI) a(n)=if(n<=1, [1,2][n+1], 1+(2^(2^(n-1))-1)*a(n-1)); /* _Joerg Arndt_, Mar 11 2013 */ %Y A162634 Cf. A010060, A000215, A085394, A085395, A081706, A161627, A161639, A074072, A074073. %K A162634 nonn %O A162634 0,2 %A A162634 _Vladimir Shevelev_, Jul 08 2009, Jul 14 2009 %E A162634 Added more terms, _Joerg Arndt_, Mar 11 2013