A332062 - OEIS

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

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OFFSET

0,1

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

Table of n, a(n) for n=0..42.

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

FORMULA

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

PROG

(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

CROSSREFS

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

Sequence in context: A118084 A232481 A232482 * A253062 A117590 A308991

Adjacent sequences: A332059 A332060 A332061 * A332063 A332064 A332065

KEYWORD

nonn

AUTHOR

M. F. Hasler, Feb 22 2020

EXTENSIONS

More terms from Jinyuan Wang, Mar 08 2020

STATUS

approved