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A273066
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Decimal expansion of the real root of x^3 - 2x + 2, negated.
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4
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1, 7, 6, 9, 2, 9, 2, 3, 5, 4, 2, 3, 8, 6, 3, 1, 4, 1, 5, 2, 4, 0, 4, 0, 9, 4, 6, 4, 3, 3, 5, 0, 3, 3, 4, 9, 2, 6, 7, 0, 5, 5, 3, 0, 4, 5, 8, 9, 8, 8, 5, 7, 0, 0, 4, 2, 3, 3, 1, 0, 6, 1, 3, 0, 4, 0, 2, 6, 7, 3, 8, 1, 7, 3, 5, 0, 6, 6, 8, 3, 2, 9, 0, 6, 8, 7, 4, 1, 2, 2, 1, 4, 9, 4, 4, 5, 4, 8, 1, 8, 1, 2, 7, 1, 6
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OFFSET
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1,2
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COMMENTS
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The roots of x^3 + A273065*x^2 - A273066*x + A273067 are A273065, -A273066, and A273067. The only other real, cubic, monic polynomial with nonzero constant term and equal coefficients and roots when ignoring the leading coefficient is x^3 + x^2 - x - 1 (per the Math Overflow link).
The other two roots of x^3 - 2*x - 2 are w1*(1 + (1/9)*sqrt(57))^(1/3) + w2*(1 - (1/9)*sqrt(57))^(1/3) = -0.88464617... + 0.58974280...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1. Using hyperbolic functions this is -(1/3)*sqrt(6)*(cosh((1/3)*arccosh((3/4)*sqrt(6))) - sqrt(3)*sinh((1/3)*arccosh((3/4)*sqrt(6)))*i) and its complex conjugate. - Wolfdieter Lang, Sep 13 2022
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LINKS
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FORMULA
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Equals ((9-sqrt(57))^(1/3))/(3^(2/3)) + 2/((3(9-sqrt(57)))^(1/3)) (from Wolfram Alpha).
Equals (1 + (1/9)*sqrt(57))^(1/3) + (2/3)*(1 + (1/9)*sqrt(57))^(-1/3) [compare with the above formula which uses the negative sqrt(57)].
Equals (1 + (1/9)*sqrt(57))^(1/3) + (1 - (1/9)*sqrt(57))^(1/3).
Equals (2/3)*sqrt(6)*cosh((1/3)*arccosh((3/4)*sqrt(6))). (End)
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EXAMPLE
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1.7692923542386314152404094643350334926705530458988570042331061304026738...
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MATHEMATICA
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RealDigits[Root[x^3-2x+2, 1], 10, 120][[1]] (* Harvey P. Dale, May 25 2022 *)
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PROG
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(PARI) default(realprecision, 200);
-solve(x = -1.8, -1.7, x^3 - 2*x + 2)
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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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