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A300078
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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.
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3
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OFFSET
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0,2
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COMMENTS
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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, ...
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LINKS
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MAPLE
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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);
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PROG
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(Python)
from fractions import Fraction
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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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