cp's OEIS Frontend

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.

%I A384513 #10 Jun 05 2025 22:34:10
%S A384513 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,
%T A384513 15,16,16,17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,25,26,27,27,
%U A384513 28,28,28,29,30,30,31,31,32,32,33,33,34,34,35,35,36,36,37,38,38,39,39,40
%N 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.
%C A384513 a(n)/n seems to converge to Pi/6.
%C A384513 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.
%H A384513 Luke Bennet, <a href="/A384513/b384513.txt">Table of n, a(n) for n = 1..10001</a>
%H A384513 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 A384513 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 A384513 (Python)
%o A384513 def a(n):
%o A384513     dps = 1
%o A384513     while True:
%o A384513         mpmath.iv.dps = dps
%o A384513         c = iv.mpc(iv.mpf(16) / n ** 2 + 0.375, iv.mpf(1) / n + iv.sqrt(3) / 8)
%o A384513         z = iv.mpc(0, 0)
%o A384513         counter = 0
%o A384513         while (z.real**2 + z.imag**2).b <= 4:
%o A384513             z = z ** 2 + c
%o A384513             counter += 1
%o A384513         if (z.real**2 + z.imag**2).a > 4:
%o A384513             return counter
%o A384513         dps *= 2
%Y A384513 Cf. A019673, A097486, A383750, A384509.
%K A384513 nonn
%O A384513 1,4
%A A384513 _Luke Bennet_, May 31 2025