A101801 Decimal expansion of the area of the ampersand curve.

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

Offset

1,3

Links

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

Eric Weisstein's World of Mathematics, Ampersand Curve

Example

1.06655509...

Mathematica

eq = (y^2 - x^2)(x - 1)(2x - 3) == 4 (x^2 + y^2 - 2x)^2 ;
sy = Solve[eq, y]; f1[x_] = y /. sy[[2]]; f2[x_] = y /. sy[[4]]; x1 = x /.    FindRoot[f1'[x] == 1, {x, 31/21}, WorkingPrecision -> 120 ]; y1 = y /. Solve[eq /. x -> x1][[3]]; y2 =  y /. Solve[eq /. x -> x1][[4]]; sx = Solve[eq, x]; g1[y_] = x /. sx[[1]]; g2[y_] = Simplify[x /. sx[[4]], y1 < y < y2]; ni[a_, b_] := NIntegrate[a, b, WorkingPrecision -> 120]; i1 = ni[Simplify[-g1[y], 0 < y < Sqrt[3]/2] , {y, 0, Sqrt[3]/2}]; i2 = ni[f2[x] - f1[x], {x, 0, 1}]; i3 = ni[f2[x] - f1[x], {x, 1, x1}]; i4 = ni[g2[y] - x1 , {y, y1, y2}]; Take[RealDigits[2 (i1 + i2 + i3 + i4)][[1]], 105]
(* Jean-François Alcover, Aug 03 2011 *)

Crossrefs

Sequence in context: A246184 A197478 A260713 * A350362 A351120 A199664

Adjacent sequences: A101798 A101799 A101800 * A101802 A101803 A101804

Keyword

nonn,cons

Author

Eric W. Weisstein, Dec 16 2004

Status

approved