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A097486 A relationship between Pi and the Mandelbrot set. a(n) = number of iterations of z^2 + c that c-values -.75 + x*i go through before escaping, where x = 10^(-n). Lim_{n->inf} a(n) * x = Pi. 2
3, 33, 315, 3143, 31417, 314160, 3141593, 31415927, 314159266, 3141592655, 31415926537, 314159265359, 3141592653591 (list; graph; refs; listen; history; text; internal format)



-.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.

Calculating 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.


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.


Table of n, a(n) for n=0..12.

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 [From Hans Havermann, Feb 12 2010]

Aaron Klebanoff, Pi in the Mandelbrot set (proof)

Robert Munafo, Seahorse Valley [From Robert Munafo, Jan 25 2010]


$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 *)


(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

(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


Sequence in context: A207323 A135697 A226508 * A121515 A221883 A002277

Adjacent sequences:  A097483 A097484 A097485 * A097487 A097488 A097489




Gerald McGarvey, Sep 19 2004


Links corrected by Gerald McGarvey, Dec 16 2009

Corrected and extended by Robert Munafo, Jan 25 2010



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Last modified March 23 04:20 EDT 2017. Contains 283902 sequences.