A299415 -id:A299415 - cp's OEIS frontend

A097486 A relationship between Pi and the Mandelbrot set. a(n) = number of iterations of z^2 + c that c-values -0.75 + xi go through before escaping, where x = 10^(-n). Lim_{n->inf} a(n) x = Pi.

3, 33, 315, 3143, 31417, 314160, 3141593, 31415927, 314159266, 3141592655, 31415926537, 314159265359, 3141592653591

Offset: 0

Gerald McGarvey, Sep 19 2004

-0.75 + 0*i is the neck of the Mandelbrot set.
a(n) is an approximation to Pi*10^n. If you substitute "1/K" in place of "0.1" in the algorithm, the resulting sequence will approximate Pi*K^n. If expressed in base K, the sequence terms will then have digits similar to the digits of Pi in base K.
Calculation of this sequence is subject to roundoff errors. In PARI/GP and in C++ using a quad-precision library, the value of A(7) is 31415927, not 31415928 as was originally recorded in this entry. - Robert Munafo, Jan 07 2010
In the PARI/GP program below, if you change "z=0" to "z=c" and "2.0" to "4.0", you get a similar sequence and in addition, A(-1)=0, which is "more aesthetically correct" given the notion that this sequence approximates Pi*10^n. However, such a modified program is NOT equivalent for positive N, it gives A097486(8)=314159267. - Robert Munafo, Jan 25 2010
Terms through a(9) verified in MAGMA by Jason Kimberley, and in Mathematica by Hans Havermann.
The difference between the terms of a(n) and A011545(n) = floor(Pi*10^n) is d(n) = 0, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, ... - Martin Renner, Feb 24 2018

Peitgen, Juergens and Saupe: Chaos and Fractals (Springer-Verlag 1992) pages 859-862.
Peitgen, Juergens and Saupe: Fractals for the Classroom (Springer-Verlag 1992) Part two, pages 431-434.

Dave Boll, Pi and the Mandelbrot set.
Boris Gourevitch, Pi et les fractales, Ensemble de Mandelbrot - Dave Boll - Gerald Edgar.
Hans Havermann, Computing pi in seahorse valley. - _Hans Havermann_, Feb 12 2010
Aaron Klebanoff, Pi in the Mandelbrot set (proof).
Robert Munafo, Seahorse Valley. - _Robert Munafo_, Jan 25 2010

Cf. A011545, A299415, A300078.

A097486:=function(n) c:=10^-n*Sqrt(-1)-3/4; z:=0; a:=0; while Modulus(z)lt 2 do z:=z^2+c; a+:=1; end while; return a; end function; // Jason Kimberley

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:=-0.75+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); # Martin Renner, Feb 24 2018

$MinPrecision = 128; Do[c = SetPrecision[.1^n * I - .75, 128]; z = 0; a = 0; While[Abs[z] < 2, z = z^2 + c; a++ ]; Print[a], {n, 0, 8}] (* Hans Havermann, Oct 20 2010 *)

A097486(n)=local(a,c,z);c=0.1^n*I-0.75;z=0;a=0;while(abs(z)<2.0,{z=z^2+c;a=a+1});a \\ Robert Munafo, Jan 25 2010

Links corrected by Gerald McGarvey, Dec 16 2009
Corrected and extended by Robert Munafo, Jan 25 2010
Name corrected by Martin Renner, Feb 24 2018

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.

Original entry on oeis.org

1, 18, 159, 1586, 15731, 157085, 1570800, 15707976

Offset: 0

Martin Renner, Feb 24 2018

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, ...

Gerald Edgar, Pi and the Mandelbrot set. (The Ohio State University.)
Boris Gourévitch, 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.

Cf. A097486, A299415, A300077.

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);

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

A332062 Number of iterations of z -> z^2 + 1/4 + 1/2^n until z > 2, starting with z = 0.

Original entry on oeis.org

2, 3, 5, 7, 11, 16, 23, 34, 48, 69, 99, 140, 199, 282, 400, 567, 802, 1135, 1607, 2273, 3215, 4548, 6432, 9097, 12866, 18196, 25734, 36394, 51470, 72790, 102942, 145582, 205885, 291167, 411773, 582336, 823548, 1164673, 1647097, 2329348, 3294197, 4658698, 6588395

Offset: 0

M. F. Hasler, Feb 22 2020

nonn

The iterated map is of the form of the maps f_c: z -> z^2 + c used to define the Mandelbrot set as those complex c for which the trajectory of 0 under f_c will never leave the ball of radius 2.
The largest real number in the Mandelbrot set is c = 1/4, with the trajectory of 0 going to 1/2 from the left.
The number of iterations N(epsilon) to reach z > 2 for c = 1/4 + epsilon is such that N(epsilon) ~ Pi/sqrt(epsilon), see the Numberphile video.

Brady Haran and Holly Krieger, Pi and the Mandelbrot Set, Numberphile channel on YouTube, Oct. 1, 2015.

Cf. A332061 (contains this as subsequence), A299415 (variant based on the same idea, with 1/10^n instead of 1/2^n).

apply( {A332062(n)=A332061(2^n)}, [0..35]) \\ may take about a second

A332062 = lambda n: A332061(2**n) # Warning: may give incorrect result for default (double) precision for n > 40. - Giovanni Resta, Mar 08 2020

a(n) = A332061(2^n) ~ Pi*2^(n/2), asymptotically.

More terms from Jinyuan Wang, Mar 08 2020

cp's OEIS Frontend

A097486 A relationship between Pi and the Mandelbrot set. a(n) = number of iterations of z^2 + c that c-values -0.75 + xi go through before escaping, where x = 10^(-n). Lim_{n->inf} a(n) x = Pi.

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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.

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Maple

Python

A332062 Number of iterations of z -> z^2 + 1/4 + 1/2^n until z > 2, starting with z = 0.

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PARI

Python

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cp's OEIS Frontend

A097486 A relationship between Pi and the Mandelbrot set. a(n) = number of iterations of z^2 + c that c-values -0.75 + x*i go through before escaping, where x = 10^(-n). Lim_{n->inf} a(n) * x = Pi.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Extensions

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Python

A332062 Number of iterations of z -> z^2 + 1/4 + 1/2^n until z > 2, starting with z = 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Formula

Extensions

A097486 A relationship between Pi and the Mandelbrot set. a(n) = number of iterations of z^2 + c that c-values -0.75 + xi go through before escaping, where x = 10^(-n). Lim_{n->inf} a(n) x = Pi.