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A097486
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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.
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1
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3, 33, 315, 3143, 31417, 314160, 3141593, 31415927, 314159266, 3141592655, 31415926537, 314159265359, 3141592653591
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| -.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
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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
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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 [From Hans Havermann, Feb 12 2010]
Aaron Klebanoff, Pi in the Mandelbrot set (proof)
R. Munafo, Seahorse Valley [From Robert Munafo, Jan 25 2010]
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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
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PROG
| (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
(Other) (PARI/GP) 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 [From Robert Munafo, Jan 25 2010]
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CROSSREFS
| Sequence in context: A043038 A107127 A135697 * A121515 A002277 A001507
Adjacent sequences: A097483 A097484 A097485 * A097487 A097488 A097489
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KEYWORD
| nonn
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AUTHOR
| Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Sep 19 2004
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EXTENSIONS
| Links corrected by Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Dec 16 2009
Corrected and extended by Robert Munafo (mrob27(AT)gmail.com), Jan 25 2010
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