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A299415
Number of steps of iterating z -> z^2 + c with c = 1/4 + 10^(-n) to reach z > 2, starting with z = 0.
4
2, 8, 30, 97, 312, 991, 3140, 9933, 31414, 99344, 314157, 993457, 3141591, 9934586, 31415925
OFFSET
0,1
COMMENTS
A relation between Pi and the Mandelbrot set: a(n)*10(-n/2) converges to Pi.
c = 1/4 is the largest real number in the Mandelbrot set.
The difference between the terms of b(n) = floor(Pi*sqrt(10^n)) = 3, 9, 31, 99, 314, 993, 3141, 9934, 31415, 99345, 314159, 993458, ... and a(n) is d(n) = 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
REFERENCES
Heinz-Otto Peitgen; Hartmut Jürgens; Dietmar Saupe: Chaos. Bausteine der Ordnung. Berlin; Heidelberg: Springer, 1994, p. 452-456.
LINKS
Gerald Edgar, Pi and the Mandelbrot set. (The Ohio State University.)
Brady Haran and Holly Krieger, Pi and the Mandelbrot Set, Numberphile channel on YouTube, Oct. 1, 2015.
Aaron Klebanoff, Pi in the Mandelbrot Set, Fractals 9 (2001), nr. 4, p. 393-402.
MAPLE
Digits:=10^3:
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:=0.25+epsilon:
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..11);
MATHEMATICA
digits = 10^3;
f[z_, c_, k_] := f[z, c, k] = f[z, c, k-1]^2 + c;
a[n_] := Module[{epsilon = 10^-n, c, k}, c = N[1/4 + epsilon, digits]; f[0, c, 0] = 0; For[k = 1, True, k++, If[Abs[f[0, c, k]] > 2, Break[]]]; k];
a /@ Range[0, 11] (* Jean-François Alcover, Nov 05 2020, after Maple *)
PROG
(PARI) apply( {A299415(n)=A332061(10^n)}, [0..12]) \\ a(12) may take about a second to compute. - M. F. Hasler, Feb 22 2020
(Python) A299415 = lambda n: A332061(10**n) \\ Warning: may give incorrect result for default (double) precision for n >= 12. - M. F. Hasler, Feb 22 2020
CROSSREFS
Cf. A332061, A332062 (same with epsilon = 1/n resp. 1/2^n).
Sequence in context: A052437 A131318 A010749 * A364078 A373904 A230701
KEYWORD
nonn,more
AUTHOR
Martin Renner, Feb 21 2018
EXTENSIONS
Edited and extended to a(15) by M. F. Hasler, Feb 22 2020
STATUS
approved