A193670 Decimal expansion of Ampersand curve length.

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

Offset

1,1

Links

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

Eric Weisstein's World of Mathematics, Ampersand Curve

Example

9.1975597439...

Mathematica

eq = (y^2 - x^2)(x - 1)(2 x - 3) == 4(x^2 + y^2 - 2 x)^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_] = x /. sx[[4]] // Simplify[#, y1 < y < y2] &; ni[a_, b_] := NIntegrate[a, b, WorkingPrecision -> 120]; i1 = ni[Sqrt[1+f1'[x]^2], {x, 0, 1}] + ni[Sqrt[1+f1'[x]^2], {x, 1, x1}]; i2 = ni[Sqrt[1+f2'[x]^2], {x, 0, 1}] + ni[Sqrt[1+f2'[x]^2], {x, 1, x1}]; i3 = ni[Sqrt[1+g1'[y]^2], {y, 0, Sqrt[3]/2}]; i4 = ni[Sqrt[1+g2'[y]^2], {y, y1, y2}]; Take[RealDigits[2(i1+i2+i3+i4)][[1]], 105]

Crossrefs

Cf. A101801 (area).

Sequence in context: A176518 A154697 A187368 * A309614 A154220 A133919

Adjacent sequences: A193667 A193668 A193669 * A193671 A193672 A193673

Keyword

nonn,cons

Author

Jean-François Alcover, Aug 02 2011

Status

approved