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 A193670 Decimal expansion of Ampersand curve length. 0

%I

%S 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,

%T 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,

%U 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

%N Decimal expansion of Ampersand curve length.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AmpersandCurve.html">Ampersand Curve</a>

%e 9.1975597439...

%t 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]

%Y Cf. A101801 (area).

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Aug 02 2011

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Last modified September 28 06:57 EDT 2021. Contains 347703 sequences. (Running on oeis4.)