A204619 Decimal expansion of arc length of the Keratoid Cusp curve (loop arc length).

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

Offset

0,1

Links

Table of n, a(n) for n=0..104.

Eric Weisstein's World of Mathematics, Keratoid Cusp

Example

0.5100949418...

Mathematica

eq = y^2 == x^2*y + x^5; f1[x_] = y /. Solve[eq, y][[1]]; f2[x_] = y /. Solve[eq, y][[2]]; x1 = (x - 1/4)/2 /. Solve[f2'[x] == 0][[1]]; y1 = f1[x1]; y2 = f2[x1]; g[y_] = x /. Solve[eq, x][[1]]; gp[y_] := (2*y - g[y]^2)/(2*y*g[y] + 5*g[y]^4); ni[a_, b_] := NIntegrate[a, b, WorkingPrecision -> 120]; i1 = ni[Sqrt[1 + f1'[x]^2], {x, x1, 0}]; i2 = ni[Sqrt[1 + f2'[x]^2], {x, x1, 0}]; i3 = ni[Sqrt[1 + gp[y]^2], {y, y1, y2}]; Take[RealDigits[i1 + i2 + i3][[1]], 105] (* Jean-François Alcover, Jan 17 2012 *)

Crossrefs

Sequence in context: A343016 A058177 A345373 * A228077 A204170 A283784

Adjacent sequences: A204616 A204617 A204618 * A204620 A204621 A204622

Keyword

nonn,cons

Author

Jean-François Alcover, Jan 17 2012

Extensions

Offset corrected by Rick L. Shepherd, Jan 05 2014

Status

approved