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

2, 3, 5, 7, 11, 16, 23, 34, 48, 69, 99, 140, 199, 282, 400, 567, 802, 1135, 1607, 2273, 3215, 4548, 6432, 9097, 12866, 18196, 25734, 36394, 51470, 72790, 102942, 145582, 205885, 291167, 411773, 582336, 823548, 1164673, 1647097, 2329348, 3294197, 4658698, 6588395

Offset: 0

Author

M. F. Hasler, Feb 22 2020

Keywords

• nonn

Comments

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.

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

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.

Links

• Brady Haran and Holly Krieger, Pi and the Mandelbrot Set, Numberphile channel on YouTube, Oct. 1, 2015.

Crossrefs

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

Programs

PARI

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

Python

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

Formula

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

Extensions

More terms from Jinyuan Wang, Mar 08 2020