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.

%I A383750 #24 Jun 05 2025 23:46:33
%S A383750 1,2,4,6,8,10,11,13,15,17,19,20,22,24,26,28,29,31,33,35,37,38,40,42,
%T A383750 44,46,47,49,51,53,55,57,58,60,62,64,66,68,69,71,73,75,77,78,80,82,84,
%U A383750 86,87,89,91,93,95,96,98,100,102,104,105,107,109,111,113,115,116,118,120
%N 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.
%C A383750 a(n)/n appears to converge to Pi/sqrt(3).
%C A383750 a(n) counts the escape time of points outside the Mandelbrot set that converge to the Mandelbrot set's 1/3 period bulb.
%H A383750 Luke Bennet, <a href="/A383750/b383750.txt">Table of n, a(n) for n = 1..10001</a>
%H A383750 Thies Brockmöller, Oscar Scherz, and Nedim Srkalović, <a href="https://arxiv.org/abs/2505.07138">Pi in the Mandelbrot set everywhere</a>, arXiv preprint arXiv:2505.07138 [math.DS], 2025.
%H A383750 Aaron Klebanoff, <a href="https://www.doc.ic.ac.uk/~jb/teaching/jmc/pi-in-mandelbrot.pdf">Pi in the Mandelbrot Set</a>, Fractals 9 (2001), nr. 4, p. 393-402.
%o A383750 (Python)
%o A383750 import mpmath
%o A383750 from mpmath import iv
%o A383750 def a(n):
%o A383750     dps = 1
%o A383750     while True:
%o A383750         mpmath.iv.dps = dps
%o A383750         real_part = iv.mpf(1) / n - iv.mpf('0.125')
%o A383750         imag_part = iv.mpf(2) / (n ** 2) + 3 * iv.sqrt(3) / 8
%o A383750         c = iv.mpc(real_part, imag_part)
%o A383750         z = iv.mpc(0, 0)
%o A383750         counter = 0
%o A383750         while (z.real**2 + z.imag**2).b <= 4:
%o A383750             z = z ** 2 + c
%o A383750             counter += 1
%o A383750         if (z.real**2 + z.imag**2).a > 4:
%o A383750             return counter
%o A383750         dps *= 2
%Y A383750 Cf. A093602, A097486, A384509, A384513
%K A383750 nonn
%O A383750 1,2
%A A383750 _Luke Bennet_, May 08 2025