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A300078 Number of steps of iterating 0 under z^2 + c before escaping, i.e., abs(z^2 + c) > 2, with c = -5/4 - epsilon^2 + epsilon*i, where epsilon = 10^(-n) and i^2 = -1.

%I A300078 #14 Mar 03 2018 17:36:03
%S A300078 1,18,159,1586,15731,157085,1570800,15707976
%N A300078 Number of steps of iterating 0 under z^2 + c before escaping, i.e., abs(z^2 + c) > 2, with c = -5/4 - epsilon^2 + epsilon*i, where epsilon = 10^(-n) and i^2 = -1.
%C A300078 A relation between Pi and the Mandelbrot set: 2*a(n)*epsilon converges to Pi.
%C A300078 c = -5/4 - epsilon^2 + epsilon*i is a parabolic route into the point c = -5/4, the second neck of the Mandelbrot set.
%C A300078 The difference between the terms of a(n) and A300077(n) = floor(1/2*Pi*10^n) is d(n) = 0, 3, 2, 16, 24, 6, 4, 13, ...
%H A300078 Gerald Edgar, <a href="https://people.math.osu.edu/edgar.2/piand.html">Pi and the Mandelbrot set.</a> (The Ohio State University.)
%H A300078 Boris Gourévitch, <a href="http://www.pi314.net/eng/mandelbrot.php">Pi and fractal sets. The Mandelbrot set -- Dave Boll -- Gerald Edgar.</a> (The World of Pi.)
%H A300078 Aaron Klebanoff, <a href="https://pdfs.semanticscholar.org/dbed/13dae724fed20356b81be91c63fc13b1e1b8.pdf">Pi in the Mandelbrot Set.</a> In: Fractals 9 (2001), nr. 4, p. 393-402.
%p A300078 Digits:=2^8:
%p A300078 f:=proc(z, c, k) option remember;
%p A300078   f(z, c, k-1)^2+c;
%p A300078 end;
%p A300078 a:=proc(n)
%p A300078 local epsilon, c, k;
%p A300078   epsilon:=10.^(-n):
%p A300078   c:=-1.25-epsilon^2+epsilon*I:
%p A300078   f(0, c, 0):=0:
%p A300078   for k do
%p A300078     if abs(f(0, c, k))>2 then
%p A300078       break;
%p A300078     fi;
%p A300078   od:
%p A300078   return(k);
%p A300078 end;
%p A300078 seq(a(n), n=0..7);
%o A300078 (Python)
%o A300078 from fractions import Fraction
%o A300078 def A300078(n):
%o A300078     zr, zc, c = Fraction(0,1), Fraction(0,1), 0
%o A300078     cr, cc = Fraction(-5,4)-Fraction(1,10**(2*n)), Fraction(1,10**n)
%o A300078     zr2, zc2 = zr**2, zc**2
%o A300078     while zr2 + zc2 <= 4:
%o A300078         zr, zc = zr2 - zc2 + cr, 2*zr*zc + cc
%o A300078         zr2, zc2 = zr**2, zc**2
%o A300078         c += 1
%o A300078     return c # _Chai Wah Wu_, Mar 03 2018
%Y A300078 Cf. A097486, A299415, A300077.
%K A300078 nonn,more
%O A300078 0,2
%A A300078 _Martin Renner_, Feb 24 2018