A332061 -id:A332061 - cp's OEIS frontend

A299415 Number of steps of iterating z -> z^2 + c with c = 1/4 + 10^(-n) to reach z > 2, starting with z = 0.

2, 8, 30, 97, 312, 991, 3140, 9933, 31414, 99344, 314157, 993457, 3141591, 9934586, 31415925, 99345881, 314159263, 993458825, 3141592652, 9934588264

Offset: 0

Martin Renner, Feb 21 2018

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, ...

Heinz-Otto Peitgen; Hartmut Jürgens; Dietmar Saupe: Chaos. Bausteine der Ordnung. Berlin; Heidelberg: Springer, 1994, p. 452-456.

Gerald Edgar, Pi and the Mandelbrot set. (The Ohio State University.)
Boris Gourévitch, Pi and fractal sets. The Mandelbrot set -- Dave Boll -- Gerald Edgar. (The World of Pi.)
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.

Cf. A011545, A097486, A300078.

Cf. A332061, A332062 (same with epsilon = 1/n resp. 1/2^n).

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);

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 *)

apply( {A299415(n)=A332061(10^n)}, [0..12]) \\ a(12) may take about a second to compute. -  M. F. Hasler, Feb 22 2020

A299415 = lambda n: A332061(10**n) # Warning: may give incorrect result for default (double) precision for n >= 12. -  M. F. Hasler, Feb 22 2020

Edited and extended to a(14) by M. F. Hasler, Feb 22 2020

a(15)-a(19) from Bill McEachen, Aug 10 2025

A332062 Number of iterations of z -> z^2 + 1/4 + 1/2^n until z > 2, starting with z = 0.

Original entry on oeis.org

2, 3, 5, 7, 11, 16, 23, 34, 48, 69, 99, 140, 199, 282, 400, 567, 802, 1135, 1607, 2273, 3215, 4548, 6432, 9097, 12866, 18196, 25734, 36394, 51470, 72790, 102942, 145582, 205885, 291167, 411773, 582336, 823548, 1164673, 1647097, 2329348, 3294197, 4658698, 6588395

Offset: 0

M. F. Hasler, Feb 22 2020

nonn

The iterated map is of the form of the maps f_c: z -> z^2 + c used to define the Mandelbrot set as those complex c for which the trajectory of 0 under f_c will never leave the ball of radius 2.
The largest real number in the Mandelbrot set is c = 1/4, with the trajectory of 0 going to 1/2 from the left.
The number of iterations N(epsilon) to reach z > 2 for c = 1/4 + epsilon is such that N(epsilon) ~ Pi/sqrt(epsilon), see the Numberphile video.

Brady Haran and Holly Krieger, Pi and the Mandelbrot Set, Numberphile channel on YouTube, Oct. 1, 2015.

Cf. A332061 (contains this as subsequence), A299415 (variant based on the same idea, with 1/10^n instead of 1/2^n).

apply( {A332062(n)=A332061(2^n)}, [0..35]) \\ may take about a second

A332062 = lambda n: A332061(2**n) # Warning: may give incorrect result for default (double) precision for n > 40. - Giovanni Resta, Mar 08 2020

a(n) = A332061(2^n) ~ Pi*2^(n/2), asymptotically.

More terms from Jinyuan Wang, Mar 08 2020

cp's OEIS Frontend

A299415 Number of steps of iterating z -> z^2 + c with c = 1/4 + 10^(-n) to reach z > 2, starting with z = 0.

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