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A086694 A run of 2^n 1's followed by a run of 2^n 0's, for n=0, 1, 2, ...

%I A086694 #39 Jan 12 2025 03:49:04
%S A086694 1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,1,1,
%T A086694 1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,
%U A086694 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0
%N A086694 A run of 2^n 1's followed by a run of 2^n 0's, for n=0, 1, 2, ...
%C A086694 First differences of A006165 and, likely, of A078881.
%H A086694 Robert Israel, <a href="/A086694/b086694.txt">Table of n, a(n) for n = 1..10000</a>
%H A086694 Ralf Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A086694 Ralf Stephan, <a href="/A079944/a079944.ps">Table of generating functions</a>
%F A086694 a(n) = 1-A079944(n-1) = 2-A079882(n-1) = A080791(n+1)-A083661(n+1).
%F A086694 a(n) = 1 - floor(log_2(4*(n+1)/3)) + floor(log_2(n+1)).
%F A086694 a(1) = 1, a(2) = 0, a(2n+1) = a(n), a(2n) = a(n-1).
%F A086694 G.f.: Sum_{k>=1} (x^(2^k)-x^(3*2^(k-1)))/(x-x^2). - _Robert Israel_, Jul 27 2017
%F A086694 G.f.: g(x) = (1/(1 - x))*( Sum_{n >= 1} x^(2^n-1)*(1 - x^2^(n-1)) ). Functional equation: g(x) = x + x*(1+x)*g(x^2). - _Wolfgang Hintze_, Aug 05 2017
%p A086694 seq(op([1$(2^n),0$(2^n)]),n=0..6); # _Robert Israel_, Jul 27 2017
%t A086694 Table[{PadRight[{},2^n,1],PadRight[{},2^n,0]},{n,0,5}]//Flatten (* _Harvey P. Dale_, May 29 2017 *)
%t A086694 Table[{Array[1&,2^n],Array[0&,2^n]},{n,0,5}]//Flatten (* _Wolfgang Hintze_, Jul 27 2017 *)
%o A086694 (PARI) a(n)=if(n<3,if(n<2,1,0),if(n%2==0,a(n/2-1),a((n-1)/2)))
%Y A086694 Cf. A005942, A079944.
%K A086694 nonn,easy
%O A086694 1,1
%A A086694 _Ralf Stephan_, Sep 12 2003