A384513 a(n) = number of iterations of z -> z^2 + c(n) with c(n) = 16/(n^2) + (1/n)*i + 3/8 + (sqrt(3)/8)*i to reach |z| > 2, starting with z = 0.

1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 6, 7, 7, 8, 8, 9, 9, 10, 10, 10, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 28, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40

Offset: 1

Author

Luke Bennet, May 31 2025

Keywords

• nonn

Comments

a(n)/n seems to converge to Pi/6.

a(n) counts the escape time of points outside the Mandelbrot set that converge to the Mandelbrot set's 1/6 period bulb. This is a proven fact and was the motivation for creating the sequence.

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. A019673, A097486, A383750, A384509.

Programs

Python

def a(n):
    dps = 1
    while True:
        mpmath.iv.dps = dps
        c = iv.mpc(iv.mpf(16) / n ** 2 + 0.375, iv.mpf(1) / n + iv.sqrt(3) / 8)
        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