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A332062 Number of iterations of z -> z^2 + 1/4 + 1/2^n until z > 2, starting with z = 0. 2

%I

%S 2,3,5,7,11,16,23,34,48,69,99,140,199,282,400,567,802,1135,1607,2273,

%T 3215,4548,6432,9097,12866,18196,25734,36394,51470,72790,102942,

%U 145582,205885,291167,411773,582336,823548,1164673,1647097,2329348,3294197,4658698,6588395

%N Number of iterations of z -> z^2 + 1/4 + 1/2^n until z > 2, starting with z = 0.

%C The iterated map is of the form of the maps f_c: z -> z^2 + c used to define the Mandelbrot set as those complex c for which the trajectory of 0 under f_c will never leave the ball of radius 2.

%C The largest real number in the Mandelbrot set is c = 1/4, with the trajectory of 0 going to 1/2 from the left.

%C The number of iterations N(epsilon) to reach z > 2 for c = 1/4 + epsilon is such that N(epsilon) ~ Pi/sqrt(epsilon), see the Numberphile video.

%H Brady Haran and Holly Krieger, <a href="https://youtu.be/d0vY0CKYhPY">Pi and the Mandelbrot Set</a>, Numberphile channel on YouTube, Oct. 1, 2015.

%F a(n) = A332061(2^n) ~ Pi*2^(n/2), asymptotically.

%o (PARI) apply( {A332062(n)=A332061(2^n)}, [0..35]) \\ may take about a second

%o (Python) A332062 = lambda n: A332061(2**n) # Warning: may give incorrect result for default (double) precision for n > 40. - _Giovanni Resta_, Mar 08 2020

%Y Cf. A332061 (contains this as subsequence), A299415 (variant based on the same idea, with 1/10^n instead of 1/2^n).

%K nonn

%O 0,1

%A _M. F. Hasler_, Feb 22 2020

%E More terms from _Jinyuan Wang_, Mar 08 2020

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Last modified July 4 22:15 EDT 2020. Contains 335449 sequences. (Running on oeis4.)