

A097486


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


4



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

0,1


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 quadprecision 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 (SpringerVerlag 1992) pages 859862.
Peitgen, Juergens and Saupe: Fractals for the Classroom (SpringerVerlag 1992) Part two, pages 431434.


LINKS

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]


MAPLE

Digits:=2^8:
f:=proc(z, c, k) option remember;
f(z, c, k1)^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 *)


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
(PARI) A097486(n)=local(a, c, z); c=0.1^n*I0.75; z=0; a=0; while(abs(z)<2.0, {z=z^2+c; a=a+1}); a \\ Robert Munafo, Jan 25 2010


CROSSREFS

Cf. A011545, A299415, A300078.
Sequence in context: A207323 A135697 A226508 * A121515 A221883 A002277
Adjacent sequences: A097483 A097484 A097485 * A097487 A097488 A097489


KEYWORD

nonn,more


AUTHOR

Gerald McGarvey, Sep 19 2004


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


STATUS

approved



