A193626 Decimal expansion of bicuspid curve length.

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

Offset

1,1

Links

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

Eric Weisstein's World of Mathematics, Bicuspid Curve.

Example

9.861772942...

Mathematica

f[x_, y_] = (x^2 - 1)*(x - 1)^2 + (y^2 - 1)^2; sy = Solve[f[x, y] == 0, y]; sx = Solve[f[x, y] == 0, x]; s = Solve[f[x, -x + 1/2] == 0, x] ; f1[x_] = y /. sy[[4, 1]]; f2[x_] = y /. sy[[2, 1]]; g1[y_] = x /. sx[[3, 1]]; g2[y_] = x /. sx[[4, 1]]; x2 = x /. s[[3]]; y2 = f1[x2]; x6 = x /. s[[4]]; y6 = f2[x6]; ni[a_, b_] := NIntegrate[a, b, WorkingPrecision -> 120]; ds1 = Sqrt[1 + f1'[x]^2] // Simplify; p1 = ni[ds1, {x, x2, 1} ] ; ds2 = Sqrt[1 + g1'[y]^2]; p2 = ni[ds2, {y, 0, y2}] ; ds3 = Sqrt[1 + g2'[y]^2]; p3 = ni[ds3, {y, 0, y6}] ; ds4 = Sqrt[1 + f2'[x]^2] // Simplify; p4 = ni[ds4, {x, x6, 1}] ; p = 2*(p1 + p2 + p3 + p4) ; Take[RealDigits[p][[1]], 105]

Crossrefs

Cf. A193625 (area)

Sequence in context: A059069 A084660 A002391 * A316600 A087044 A246168

Adjacent sequences: A193623 A193624 A193625 * A193627 A193628 A193629

Keyword

nonn,cons

Author

Jean-François Alcover, Aug 01 2011

Status

approved