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.

%I A384509 #17 Jun 05 2025 23:38:08
%S A384509 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,
%T A384509 28,29,31,32,33,34,35,36,37,38,39,40,42,43,44,45,46,47,48,49,51,51,53,
%U A384509 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
%N 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.
%C A384509 a(n)/n seems to converge to Pi/(2*sqrt(2)).
%C A384509 a(n) counts the escape time of points outside the Mandelbrot set that converge to the Mandelbrot set's 1/4 period bulb.
%H A384509 Luke Bennet, <a href="/A384509/b384509.txt">Table of n, a(n) for n = 1..10001</a>
%H A384509 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 A384509 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 A384509 (Python)
%o A384509 import mpmath
%o A384509 from mpmath import iv
%o A384509 def a(n):
%o A384509     dps = 1
%o A384509     while True:
%o A384509         mpmath.iv.dps = dps
%o A384509         c = iv.mpc(iv.mpf(5) / n + 1, iv.mpf(5) / n - 1)
%o A384509         c = c / (n * iv.sqrt(2)) + 0.25 + 0.5j
%o A384509         z = iv.mpc(0, 0)
%o A384509         counter = 0
%o A384509         while (z.real**2 + z.imag**2).b <= 4:
%o A384509             z = z ** 2 + c
%o A384509             counter += 1
%o A384509         if (z.real**2 + z.imag**2).a > 4:
%o A384509             return counter
%o A384509         dps *= 2
%o A384509 (PARI) c(n) = ((5/n+1) + (5/n-1)*I)/(n*sqrt(2)) + 1/4 + (1/2)*I;
%o A384509 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
%Y A384509 Cf. A093954, A097486, A383750, A384513.
%K A384509 nonn
%O A384509 1,2
%A A384509 _Luke Bennet_, May 31 2025