A194474 - OEIS

%I #10 Mar 31 2012 10:25:17

%S 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,

%T 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,

%U 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

%N Decimal expansion of the perimeter of the fourth Mandelbrot set lemniscate

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MandelbrotSetLemniscate.html">Mandelbrot Set Lemniscate</a>

%e 7.443644644480...

%t f[x_, y_] = ComplexExpand[#*Conjugate[#] &[c + (c + (c + c^2)^2)^2] /. c -> x + I*y] - 4 ;

%t sy = Solve[f[x, y] == 0, y]; sx = Solve[f[x, y] == 0, x];

%t f1[x_] = y /. sy[[8]]; f2[x_] = y /. sy[[4]];

%t g1[y_] = x /. sx[[1]]; g2[y_] = x /. sx[[2]];

%t x1 = -39/20; y1 = f1[x1]; x2 = -7/4; y2 = f1[x2];

%t x3 = -1; y3 = f2[x3]; x4 = -1/10; y4 = f2[x4];

%t x5 = 107/200; y5 = f1[x5]; x6 = 10703/20000; y6 = f1[x6];

%t sh = Solve[D[f[x, h[x]], x] == 0, h'[x]][[1]];

%t sg = Solve[D[f[g[y], y], y] == 0, g'[y]][[1]];

%t df1[x_] = h'[x] /. sh /. h -> f1;

%t df2[x_] = h'[x] /. sh /. h -> f2;

%t dg1[y_] = g'[y] /. sg /. g -> g1;

%t dg2[y_] = g'[y] /. sg /. g -> g2;

%t ni[a_, b_] := NIntegrate[a, b , WorkingPrecision -> 120];

%t i1 = ni[Sqrt[1 + dg1[y]^2] , {y, 0, y1}];

%t i2 = ni[Sqrt[1 + df1[x]^2], {x, x1, x2}];

%t i3 = ni[Sqrt[1 + dg1[y]^2], {y, y2, y3}];

%t i4 = ni[Sqrt[1 + df2[x]^2], {x, x3, x4}];

%t i5 = ni[Sqrt[1 + dg2[y]^2], {y, y5, y4}];

%t i6 = ni[Sqrt[1 + df1[x]^2], {x, x5, x6}];

%t i7 = ni[Sqrt[1 + dg2[y]^2], {y, 0, y6}];

%t p = 2 (i1 + i2 + i3 + i4 + i5 + i6 + i7);

%t Take[RealDigits[p][[1]], 105]

%Y Cf. A194473 (area)

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Aug 26 2011