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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. 3
 1, 18, 159, 1586, 15731, 157085, 1570800, 15707976 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A relation between Pi and the Mandelbrot set: 2*a(n)*epsilon converges to Pi. c = -5/4 - epsilon^2 + epsilon*i is a parabolic route into the point c = -5/4, the second neck of the Mandelbrot set. 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, ... LINKS Gerald Edgar, Pi and the Mandelbrot set. (The Ohio State University.) Boris GourÃ©vitch, Pi and fractal sets. The Mandelbrot set -- Dave Boll -- Gerald Edgar. (The World of Pi.) Aaron Klebanoff, Pi in the Mandelbrot Set. In: Fractals 9 (2001), nr. 4, p. 393-402. MAPLE Digits:=2^8: f:=proc(z, c, k) option remember;   f(z, c, k-1)^2+c; end; a:=proc(n) local epsilon, c, k;   epsilon:=10.^(-n):   c:=-1.25-epsilon^2+epsilon*I:   f(0, c, 0):=0:   for k do     if abs(f(0, c, k))>2 then       break;     fi;   od:   return(k); end; seq(a(n), n=0..7); PROG (Python) from fractions import Fraction def A300078(n):     zr, zc, c = Fraction(0, 1), Fraction(0, 1), 0     cr, cc = Fraction(-5, 4)-Fraction(1, 10**(2*n)), Fraction(1, 10**n)     zr2, zc2 = zr**2, zc**2     while zr2 + zc2 <= 4:         zr, zc = zr2 - zc2 + cr, 2*zr*zc + cc         zr2, zc2 = zr**2, zc**2         c += 1     return c # Chai Wah Wu, Mar 03 2018 CROSSREFS Cf. A097486, A299415, A300077. Sequence in context: A171741 A197239 A060932 * A119004 A002698 A222914 Adjacent sequences:  A300075 A300076 A300077 * A300079 A300080 A300081 KEYWORD nonn,more AUTHOR Martin Renner, Feb 24 2018 STATUS approved

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Last modified June 23 21:09 EDT 2021. Contains 345402 sequences. (Running on oeis4.)