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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.

%I A299415 #57 Aug 12 2025 20:31:04
%S A299415 2,8,30,97,312,991,3140,9933,31414,99344,314157,993457,3141591,
%T A299415 9934586,31415925,99345881,314159263,993458825,3141592652,9934588264
%N A299415 Number of steps of iterating z -> z^2 + c with c = 1/4 + 10^(-n) to reach z > 2, starting with z = 0.
%C A299415 A relation between Pi and the Mandelbrot set: a(n)*10(-n/2) converges to Pi.
%C A299415 c = 1/4 is the largest real number in the Mandelbrot set.
%C A299415 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, ...
%D A299415 Heinz-Otto Peitgen; Hartmut Jürgens; Dietmar Saupe: Chaos. Bausteine der Ordnung. Berlin; Heidelberg: Springer, 1994, p. 452-456.
%H A299415 Gerald Edgar, <a href="https://people.math.osu.edu/edgar.2/piand.html">Pi and the Mandelbrot set.</a> (The Ohio State University.)
%H A299415 Boris Gourévitch, <a href="http://www.pi314.net/eng/mandelbrot.php">Pi and fractal sets. The Mandelbrot set -- Dave Boll -- Gerald Edgar.</a> (The World of Pi.)
%H A299415 Brady Haran and Holly Krieger, <a href="https://youtu.be/d0vY0CKYhPY">Pi and the Mandelbrot Set</a>, Numberphile channel on YouTube, Oct. 1, 2015.
%H A299415 Aaron Klebanoff, <a href="https://pdfs.semanticscholar.org/dbed/13dae724fed20356b81be91c63fc13b1e1b8.pdf">Pi in the Mandelbrot Set</a>, Fractals 9 (2001), nr. 4, p. 393-402.
%p A299415 Digits:=10^3:
%p A299415 f:=proc(z,c,k) option remember;
%p A299415   f(z,c,k-1)^2+c;
%p A299415 end;
%p A299415 a:=proc(n)
%p A299415 local epsilon, c, k;
%p A299415   epsilon:=10.^(-n):
%p A299415   c:=0.25+epsilon:
%p A299415   f(0,c,0):=0:
%p A299415   for k do
%p A299415     if abs(f(0,c,k))>2 then
%p A299415       break;
%p A299415     fi;
%p A299415   od:
%p A299415   return(k);
%p A299415 end;
%p A299415 seq(a(n),n=0..11);
%t A299415 digits = 10^3;
%t A299415 f[z_, c_, k_] := f[z, c, k] = f[z, c, k-1]^2 + c;
%t A299415 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];
%t A299415 a /@ Range[0, 11] (* _Jean-François Alcover_, Nov 05 2020, after Maple *)
%o A299415 (PARI) apply( {A299415(n)=A332061(10^n)}, [0..12]) \\ a(12) may take about a second to compute. -  _M. F. Hasler_, Feb 22 2020
%o A299415 (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
%Y A299415 Cf. A011545, A097486, A300078.
%Y A299415 Cf. A332061, A332062 (same with epsilon = 1/n resp. 1/2^n).
%K A299415 nonn,more
%O A299415 0,1
%A A299415 _Martin Renner_, Feb 21 2018
%E A299415 Edited and extended to a(14) by _M. F. Hasler_, Feb 22 2020
%E A299415 a(15)-a(19) from _Bill McEachen_, Aug 10 2025