A118811 Decimal expansion of arc length of the (first) butterfly curve.

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

Offset: 1

Author

Eric W. Weisstein, Apr 30 2006

Keywords

• nonn
• cons

Examples

			9.0173569856546976918...
		

Links

• Eric Weisstein's World of Mathematics, Butterfly Curve

Crossrefs

Cf. A118292.

Programs

Mathematica

eq = y^6 == x^2-x^6; f[x_] = y /. Solve[eq, y][[2]]; g[y_] = x /. Solve[eq, x][[2]]; h[y_] = x /. Solve[eq, x][[4]]; x1 = 3/8; y1 = f[x1]; x2 = 7/8; y2 = f[x2]; ni[a_, b_] := NIntegrate[a, b, WorkingPrecision -> 120]; i1 = ni[Sqrt[1+f'[x]^2], {x, x1, x2}]; i2 = ni[Sqrt[1+g'[y]^2], {y, 0, y2}]; i3 = ni[Sqrt[1+h'[y]^2], {y, 0, y1}]; Take[RealDigits[4(i1+i2+i3)][[1]], 99](* Jean-François Alcover, Jan 19 2012 *)

PARI

4*intnum(x=0,1,sqrt(1+(x/3-x^5)^2/(x^2-x^6)^(5/3))) \\ Charles R Greathouse IV, Jan 17 2012

Extensions

Last digit corrected by Eric W. Weisstein, Jan 18 2012