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A357463
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Decimal expansion of the real root of 2*x^3 + 2*x - 1.
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1
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4, 2, 3, 8, 5, 3, 7, 9, 9, 0, 6, 9, 7, 8, 3, 2, 7, 1, 3, 7, 8, 0, 4, 0, 0, 6, 2, 6, 2, 5, 5, 1, 5, 2, 3, 3, 6, 7, 6, 3, 8, 8, 1, 9, 7, 1, 8, 5, 1, 7, 7, 5, 4, 0, 8, 2, 3, 0, 0, 8, 3, 9, 6, 8, 1, 9, 9, 5, 4, 7, 2, 8, 6, 4, 0, 7, 0, 3
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OFFSET
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0,1
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COMMENTS
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The other (complex) roots are w1*((1 + (1/9)*sqrt(129))/4)^(1/3) + ((1 - (1/9)*sqrt(129))/4)^(1/3) = -0.2119268995... + 1.0652413023...*i, and its complex conjugate, where w1 = (-1 + sqrt(3))/2 = exp((2/3)*Pi*i).
Using hyperbolic functions these roots are -(1/3)*sqrt(3)*(sinh((1/3)*arcsinh((3/4)*sqrt(3))) - sqrt(3)*cosh((1/3)*arcsinh((3/4)*sqrt(3)))*i), and its complex conjugate.
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LINKS
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FORMULA
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r = ((1 +(1/9)*sqrt(129))/4)^(1/3) - (1/3)*((1 + (1/9)*sqrt(129))/4)^(-1/3).
r = ((1 + (1/9)*sqrt(129))/4)^(1/3) + w1*((1 - (1/9)*sqrt(129))/4)^(1/3), where w1 = (-1 + sqrt(3))/2, one of the complex roots of x^3 - 1.
r = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/4)*sqrt(3))).
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EXAMPLE
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0.423853799069783271378040062625515233676388197185177540823008396819954728...
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MATHEMATICA
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RealDigits[x /. FindRoot[2*x^3 + 2*x - 1, {x, 1}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Sep 29 2022 *)
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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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