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

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

Table of n, a(n) for n=0..14.

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.

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. A011545, A097486, A300078.

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

Sequence in context: A052437 A131318 A010749 * A364078 A230701 A365693

Adjacent sequences: A299412 A299413 A299414 * A299416 A299417 A299418

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