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%I #26 Feb 13 2015 08:28:13
%S 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,
%T 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,
%U 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
%N Decimal expansion of arc length of the (first) butterfly curve.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ButterflyCurve.html">Butterfly Curve</a>
%e 9.0173569856546976918...
%t 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 *)
%o (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
%Y Cf. A118292.
%K nonn,cons
%O 1,1
%A _Eric W. Weisstein_, Apr 30 2006
%E Last digit corrected by _Eric W. Weisstein_, Jan 18 2012
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