A383750 a(n) = number of iterations of z -> z^2 + c(n) with c(n) = 1/n + (2/(n^2))*i - 1/8 + (3*sqrt(3)/8)*i to reach |z| > 2, starting with z = 0.

1, 2, 4, 6, 8, 10, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 29, 31, 33, 35, 37, 38, 40, 42, 44, 46, 47, 49, 51, 53, 55, 57, 58, 60, 62, 64, 66, 68, 69, 71, 73, 75, 77, 78, 80, 82, 84, 86, 87, 89, 91, 93, 95, 96, 98, 100, 102, 104, 105, 107, 109, 111, 113, 115, 116, 118, 120

Offset: 1

Author

Luke Bennet, May 08 2025

Keywords

• nonn

Comments

a(n)/n appears to converge to Pi/sqrt(3).

a(n) counts the escape time of points outside the Mandelbrot set that converge to the Mandelbrot set's 1/3 period bulb.

Links

• Luke Bennet, Table of n, a(n) for n = 1..10001
• Thies Brockmöller, Oscar Scherz, and Nedim Srkalović, Pi in the Mandelbrot set everywhere, arXiv preprint arXiv:2505.07138 [math.DS], 2025.
• Aaron Klebanoff, Pi in the Mandelbrot Set, Fractals 9 (2001), nr. 4, p. 393-402.

Crossrefs

Cf. A093602, A097486, A384509, A384513

Programs

Python

import mpmath
from mpmath import iv
def a(n):
    dps = 1
    while True:
        mpmath.iv.dps = dps
        real_part = iv.mpf(1) / n - iv.mpf('0.125')
        imag_part = iv.mpf(2) / (n ** 2) + 3 * iv.sqrt(3) / 8
        c = iv.mpc(real_part, imag_part)
        z = iv.mpc(0, 0)
        counter = 0
        while (z.real**2 + z.imag**2).b <= 4:
            z = z ** 2 + c
            counter += 1
        if (z.real**2 + z.imag**2).a > 4:
            return counter
        dps *= 2