OFFSET
0,1
COMMENTS
The roots of x^3 + A273065*x^2 - A273066*x + A273067 are A273065, -A273066, and A273067. See A273066, the main entry.
From Wolfdieter Lang, Sep 15 2022: (Start)
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)
FORMULA
Equals 1/A273066.
From Wolfdieter Lang, Sep 15 2022: (Start)
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)
EXAMPLE
0.565197717383639396437528013247030816098483976759553827555483810948411203...
MATHEMATICA
First[RealDigits[1/x/.N[First[Solve[x^3-2x+2==0, x]], 105]]] (* Stefano Spezia, Sep 15 2022 *)
PROG
(PARI) default(realprecision, 200);
-1/solve(x = -1.8, -1.7, x^3 - 2*x + 2)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Rick L. Shepherd, May 15 2016
STATUS
approved