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A273065
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Decimal expansion of the negative reciprocal of the real root of x^3 - 2x + 2.
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5
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5, 6, 5, 1, 9, 7, 7, 1, 7, 3, 8, 3, 6, 3, 9, 3, 9, 6, 4, 3, 7, 5, 2, 8, 0, 1, 3, 2, 4, 7, 0, 3, 0, 8, 1, 6, 0, 9, 8, 4, 8, 3, 9, 7, 6, 7, 5, 9, 5, 5, 3, 8, 2, 7, 5, 5, 5, 4, 8, 3, 8, 1, 0, 9, 4, 8, 4, 1, 1, 2, 0, 3, 3, 0, 1, 5, 7, 2, 3, 9, 4, 7, 1, 3, 3, 3, 5, 8, 7, 7, 7, 3, 9, 7, 0, 1, 1, 2, 3, 8, 4, 1, 1, 9, 0
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OFFSET
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0,1
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COMMENTS
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This equals the real root of 2*x^3 + 2*x^2 - 1, that is the real root of y^3 - (1/3)*y - 23/54, after subtracting 1/3.
The other two roots of 2*x^3 + 2*x^2 - 1 are (w1*(23/4 + (3/4)*sqrt(57))^(1/3) + w2*(23/4 - (3/4)*sqrt(57))^(1/3) - 1)/3 = -0.7825988586... + 0.5217137179...*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 these roots are -((1 + cosh((1/3)*arccosh(23/4)) + sqrt(3)*sinh((1/3)*arccosh(23/4))*i)/3) and its complex conjugate. (End)
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LINKS
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FORMULA
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Equals ((23/4 + (3/4)*sqrt(57))^(1/3) + (23/4 + (3/4)*sqrt(57))^(-1/3) - 1)/3.
Equals ((23/4 + (3/4)*sqrt(57))^(1/3) + (23/4 - (3/4)*sqrt(57))^(1/3) - 1)/3.
Equals (2*cosh((1/3)*arccosh(23/4)) - 1)/3. (End)
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EXAMPLE
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0.565197717383639396437528013247030816098483976759553827555483810948411203...
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MATHEMATICA
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First[RealDigits[1/x/.N[First[Solve[x^3-2x+2==0, x]], 105]]] (* Stefano Spezia, Sep 15 2022 *)
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PROG
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(PARI) default(realprecision, 200);
-1/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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