A384509 -id:A384509 - cp's OEIS frontend

A383750 a(n) = number of iterations of z -> z^2 + c(n) with c(n) = 1/n + (2/(n^2))i - 1/8 + (3sqrt(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

Luke Bennet, May 08 2025

nonn

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.

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.

Cf. A093602, A097486, A384509, A384513

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

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.

Original entry on oeis.org

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

Luke Bennet, May 31 2025

nonn

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.

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.

Cf. A019673, A097486, A383750, A384509.

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

cp's OEIS Frontend

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

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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.

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cp's OEIS Frontend

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.

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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.

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Python

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

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.