A383750 -id:A383750 - cp's OEIS frontend

A384509 a(n) = number of iterations of z -> z^2 + c(n) with c(n) = ((5/n+1) + (5/n-1)i)/(nsqrt(2)) + 1/4 + (1/2)*i to reach |z| > 2, starting with z = 0.

1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 51, 51, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 71, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Luke Bennet, May 31 2025

nonn

a(n)/n seems to converge to Pi/(2*sqrt(2)).

a(n) counts the escape time of points outside the Mandelbrot set that converge to the Mandelbrot set's 1/4 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. A093954, A097486, A383750, A384513.

c(n) = ((5/n+1) + (5/n-1)*I)/(n*sqrt(2)) + 1/4 + (1/2)*I;
a(n) = my(z=0, k=0, c=c(n)); while(norml2(z)<=4, z = z^2 + c; k++); k; \\ Michel Marcus, Jun 01 2025

import mpmath
from mpmath import iv
def a(n):
    dps = 1
    while True:
        mpmath.iv.dps = dps
        c = iv.mpc(iv.mpf(5) / n + 1, iv.mpf(5) / n - 1)
        c = c / (n * iv.sqrt(2)) + 0.25 + 0.5j
        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

A384509 a(n) = number of iterations of z -> z^2 + c(n) with c(n) = ((5/n+1) + (5/n-1)i)/(nsqrt(2)) + 1/4 + (1/2)*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

A384509 a(n) = number of iterations of z -> z^2 + c(n) with c(n) = ((5/n+1) + (5/n-1)*i)/(n*sqrt(2)) + 1/4 + (1/2)*i to reach |z| > 2, starting with z = 0.

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Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

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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Author

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Comments

Links

Crossrefs

Programs

Python

A384509 a(n) = number of iterations of z -> z^2 + c(n) with c(n) = ((5/n+1) + (5/n-1)i)/(nsqrt(2)) + 1/4 + (1/2)*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.