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A193670
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Decimal expansion of Ampersand curve length.
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0
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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, 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, 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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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9.1975597439...
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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