A193751 Decimal expansion of Pear curve length.

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

Offset

1,1

Comments

The Pear Curve is the third Mandelbrot set lemniscate.

Links

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

Eric Weisstein's World of Mathematics, Pear Curve

Example

7.738373624...

Mathematica

f[x_, y_] = ComplexExpand[#*Conjugate[#] &[c + (c + c^2)^2] /. c -> x + I*y] - 4; sy = Solve[f[x, y] == 0, y];
f2[x_] = y /. sy[[4]]; x2 = 3/10; y2 = f2[x2]; sx = Solve[f[x, y] == 0, x]; g1[y_] = x /. sx[[1]]; g2[y_] = x /. sx[[2]]; sg = Solve[f[g[y], y] == 0 && D[f[g[y], y], y] == 0 , g'[y]][[1]]; 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, f2[-1]} ];
i2 = ni[Sqrt[1 + f2'[x]^2], {x, -1, x2}];
i3 = ni[Sqrt[1 + dg2[y]^2], {y, 0, y2}];
Take[RealDigits[2(i1 + i2 + i3)][[1]], 105]

Crossrefs

Cf. A193750 (area)

Sequence in context: A155959 A063736 A212299 * A290565 A319263 A318386

Adjacent sequences: A193748 A193749 A193750 * A193752 A193753 A193754

Keyword

nonn,cons

Author

Jean-François Alcover, Aug 03 2011

Extensions

Corrected and extended by Jean-François Alcover, Aug 26 2011

Status

approved