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A194474 Decimal expansion of the perimeter of the fourth Mandelbrot set lemniscate 1
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 (list; constant; graph; refs; listen; history; text; internal format)
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: A153840 A198351 A245074 * A316161 A153349 A210463

Adjacent sequences:  A194471 A194472 A194473 * A194475 A194476 A194477

KEYWORD

nonn,cons

AUTHOR

Jean-François Alcover, Aug 26 2011

STATUS

approved

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Last modified September 21 09:52 EDT 2021. Contains 347597 sequences. (Running on oeis4.)