A194474 Decimal expansion of the perimeter of the fourth Mandelbrot set lemniscate

7, 4, 4, 3, 6, 4, 4, 6, 4, 4, 4, 8, 0, 0, 7, 4, 6, 2, 8, 8, 9, 0, 8, 1, 3, 4, 0, 0, 5, 8, 2, 5, 7, 6, 6, 3, 9, 3, 2, 2, 3, 1, 3, 7, 4, 4, 7, 6, 2, 5, 0, 2, 8, 1, 3, 1, 6, 5, 5, 0, 2, 9, 4, 3, 7, 2, 4, 3, 2, 1, 2, 7, 7, 6, 2, 5, 5, 1, 5, 8, 0, 5, 3, 1, 0, 7, 3, 5, 7, 3, 9, 6, 5, 6, 9, 7, 7, 5, 1, 8, 0, 4, 1, 7, 3

Offset

1,1

Links

Table of n, a(n) for n=1..105.

Eric Weisstein's World of Mathematics, Mandelbrot Set Lemniscate

Example

7.443644644480...

Mathematica

f[x_, y_] = ComplexExpand[#*Conjugate[#] &[c + (c + (c + c^2)^2)^2] /. c -> x + I*y] - 4 ;
sy = Solve[f[x, y] == 0, y]; sx = Solve[f[x, y] == 0, x];
f1[x_] = y /. sy[[8]]; f2[x_] = y /. sy[[4]];
g1[y_] = x /. sx[[1]]; g2[y_] = x /. sx[[2]];
x1 = -39/20; y1 = f1[x1]; x2 = -7/4; y2 = f1[x2];
x3 = -1; y3 = f2[x3]; x4 = -1/10; y4 = f2[x4];
x5 = 107/200; y5 = f1[x5]; x6 = 10703/20000; y6 = f1[x6];
sh = Solve[D[f[x, h[x]], x] == 0, h'[x]][[1]];
sg = Solve[D[f[g[y], y], y] == 0, g'[y]][[1]];
df1[x_] = h'[x] /. sh /. h -> f1;
df2[x_] = h'[x] /. sh /. h -> f2;
dg1[y_] = g'[y] /. sg /. g -> g1;
dg2[y_] = g'[y] /. sg /. g -> g2;
ni[a_, b_] := NIntegrate[a, b , WorkingPrecision -> 120];
i1 = ni[Sqrt[1 + dg1[y]^2] , {y, 0, y1}];
i2 = ni[Sqrt[1 + df1[x]^2], {x, x1, x2}];
i3 = ni[Sqrt[1 + dg1[y]^2], {y, y2, y3}];
i4 = ni[Sqrt[1 + df2[x]^2], {x, x3, x4}];
i5 = ni[Sqrt[1 + dg2[y]^2], {y, y5, y4}];
i6 = ni[Sqrt[1 + df1[x]^2], {x, x5, x6}];
i7 = ni[Sqrt[1 + dg2[y]^2], {y, 0, y6}];
p = 2 (i1 + i2 + i3 + i4 + i5 + i6 + i7);
Take[RealDigits[p][[1]], 105]

Crossrefs

Cf. A194473 (area)

Sequence in context: A198351 A354249 A245074 * A348736 A316161 A153349

Adjacent sequences: A194471 A194472 A194473 * A194475 A194476 A194477

Keyword

nonn,cons

Author

Jean-François Alcover, Aug 26 2011

Status

approved