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.

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

Offset: 0

Author

Gerald McGarvey, Sep 19 2004

Keywords

• nonn
• more
• changed

Comments

-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

References

• 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.

Links

• 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

Crossrefs

Cf. A011545, A299415, A300078.

Programs

Magma

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

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:=-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

Mathematica

$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 *)

PARI

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

Extensions

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