A332061 Number of iterations of z -> z^2 + 1/4 + 1/n until z >= 2, starting with z = 0.

2, 3, 4, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 24, 25, 25, 25

Offset: 1

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.

Programs

Mathematica

Table[-1 + Length@ NestWhileList[#^2 + 1/4 + 1/n &, 0, # < 2 &], {n, 73}] (* Michael De Vlieger, Feb 25 2020 *)

PARI

apply( {A332061(n,z,k)=n=.25+1/n;until(2
				

Python

def A332061(n):
    c=1/4+1/n; z=c; n=1
    while z<2: z=z**2+c; n+=1
    return n

Formula

a(n) ~ Pi*sqrt(n), asymptotically.