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 A193670 Decimal expansion of Ampersand curve length. 0
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Eric Weisstein's World of Mathematics, Ampersand Curve EXAMPLE 9.1975597439... MATHEMATICA 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] CROSSREFS Cf. A101801 (area). Sequence in context: A176518 A154697 A187368 * A309614 A154220 A133919 Adjacent sequences:  A193667 A193668 A193669 * A193671 A193672 A193673 KEYWORD nonn,cons AUTHOR Jean-François Alcover, Aug 02 2011 STATUS approved

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Last modified August 1 22:36 EDT 2021. Contains 346408 sequences. (Running on oeis4.)