A194473 Decimal expansion of the area of the fourth Mandelbrot set lemniscate

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

Offset

1,1

Links

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

Eric Weisstein's World of Mathematics, Mandelbrot Set Lemniscate

Example

3.279185833...

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];
  ni[a_, b_] := NIntegrate[a, b, WorkingPrecision -> 120];
i1 = ni[-g1[y] + g1[y1], {y, 0, y1}];
i2 = ni[f1[x], {x, x1, x2}];
i3 = ni[-g1[y] + g1[y3], {y, y2, y3}] + (x3 - x2) y2;
i4 = ni[f2[x], {x, x3, x4}] ;
i5 = ni[g2[y] - g2[y4], {y, y5, y4}] + (x5 - x4) y5 ;
i6 = ni[f1[x], {x, x5, x6}] ;
i7 = ni[ g2[y] - g2[y6], {y, 0, y6}];
area = 2 (i1 + i2 + i3 + i4 + i5 + i6 + i7);
Take[RealDigits[area][[1]], 105]

Crossrefs

Cf. A194474 (perimeter)

Sequence in context: A245581 A054170 A106167 * A091913 A212285 A192789

Adjacent sequences: A194470 A194471 A194472 * A194474 A194475 A194476

Keyword

nonn,cons

Author

Jean-François Alcover, Aug 26 2011

Status

approved