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A030300 Runs have lengths 2^n, n >= 0.

%I A030300 #57 Jan 30 2023 18:56:20
%S A030300 1,0,0,1,1,1,1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,
%T A030300 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,
%U A030300 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A030300 Runs have lengths 2^n, n >= 0.
%C A030300 An example of a sequence with property that the fraction of 1's in the first n terms does not converge to a limit. - _N. J. A. Sloane_, Sep 24 2007
%C A030300 Image, under the coding sending a,d,e -> 1 and b,c -> 0, of the fixed point, starting with a, of the morphism a -> ab, b -> cd, c -> ee, d -> eb, e -> cc. - _Jeffrey Shallit_, May 14 2016
%C A030300 This sequence taken as digits of a base-b fraction is g(1/b) = Sum_{n>=1} a(n)/b^n = b/(b-1) * Sum_{k>=0} (-1)^k/b^(2^k) per the generating function below.  With initial 0, it is binary expansion .01001111 = A275975.  With initial 0 and digits 2*a(n), it is ternary expansion .02002222 = A160386.  These and in general g(1/b) for any integer b>=2 are among forms which Kempner showed are transcendental. - _Kevin Ryde_, Sep 07 2019
%H A030300 Antti Karttunen, <a href="/A030300/b030300.txt">Table of n, a(n) for n = 1..65537</a>
%H A030300 Aubrey J. Kempner, <a href="http://www.jstor.org/stable/1988833">On Transcendental Numbers</a>, Transactions of the American Mathematical Society, volume 17, number 4, October 1916, pages 476-482.
%H A030300 Kevin Ryde, <a href="http://oeis.org/plot2a?name1=A079947&amp;name2=A000027&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=false&amp;drawlines=true">Plot of A079947(n)/n</a>, illustrating proportion of 1s in the first n terms here does not converge (but oscillates with rises and falls by hyperbolas)
%H A030300 Ralf Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A030300 Ralf Stephan, <a href="/A079944/a079944.ps">Table of generating functions</a>
%H A030300 Ralf Stephan, <a href="https://arxiv.org/abs/math/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>, arXiv:math/0307027 [math.CO], 2003.
%H A030300 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A030300 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F A030300 a(n) = A065359(n) + A083905(n).
%F A030300 a(n) = (1/2)*(1+(-1)^floor(log_2(n))). - _Benoit Cloitre_, Feb 22 2003
%F A030300 G.f.: 1/(1-x) * Sum_{k>=0} (-1)^k*x^2^k. - _Ralf Stephan_, Jul 12 2003
%F A030300 a(n) = 1 - a(floor(n/2)). - _Vladeta Jovovic_, Aug 04 2003
%F A030300 a(n) = A115253(2n, n) mod 2. - _Paul Barry_, Jan 18 2006
%F A030300 a(n) = 1 - A030301(n). - _Antti Karttunen_, Oct 10 2017
%p A030300 f0 := n->[seq(0,i=1..2^n)]; f1 := n->[seq(1,i=1..2^n)]; s := []; for i from 0 to 4 do s := [op(s), op(f1(2*i)), op(f0(2*i+1))]; od: A030300 := s;
%t A030300 nMax = 6; Table[1 - Mod[n, 2], {n, 0, nMax}, {2^n}] // Flatten (* _Jean-François Alcover_, Oct 20 2016 *)
%o A030300 (PARI) a(n) = if(n, !(logint(n,2)%2)); /* _Kevin Ryde_, Aug 02 2019 */
%o A030300 (Python)
%o A030300 def A030300(n): return n.bit_length()&1 # _Chai Wah Wu_, Jan 30 2023
%Y A030300 Cf. A030301. Partial sums give A079947.
%Y A030300 Cf. A065359, A083905.
%Y A030300 Characteristic function of A053738.
%K A030300 nonn,base,easy
%O A030300 1,1
%A A030300 _Jean-Paul Delahaye_