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

Author

Martin Renner, Feb 21 2018

Keywords

• nonn
• more

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.)
• 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.

Crossrefs

Cf. A011545, A097486, A300078.

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

Programs

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 *)

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

Extensions

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

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