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A030300
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Runs have lengths 2^n, n >= 0.
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16
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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, 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, 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
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OFFSET
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1,1
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COMMENTS
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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
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
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
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LINKS
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Aubrey J. Kempner, On Transcendental Numbers, Transactions of the American Mathematical Society, volume 17, number 4, October 1916, pages 476-482.
Kevin Ryde, Plot of A079947(n)/n, illustrating proportion of 1s in the first n terms here does not converge (but oscillates with rises and falls by hyperbolas)
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FORMULA
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G.f.: 1/(1-x) * Sum_{k>=0} (-1)^k*x^2^k. - Ralf Stephan, Jul 12 2003
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MAPLE
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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;
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MATHEMATICA
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PROG
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(PARI) a(n) = if(n, !(logint(n, 2)%2)); /* Kevin Ryde, Aug 02 2019 */
(Python)
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CROSSREFS
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Characteristic function of A053738.
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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