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%I #23 Sep 20 2022 04:58:18
%S 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,
%T 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,
%U 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
%N Decimal expansion of the real root of x^3 - 2x + 2, negated.
%C 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).
%C 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
%H A. J. Di Scala and O. Macia, <a href="http://arXiv.org/abs/0904.0133">Finiteness of Ulam Polynomials</a>, arXiv:0904.0133 [math.AG], 2009.
%H R. Stanley, R. Israel et al., <a href="http://mathoverflow.net/questions/216346">Math Overflow: Which polynomial's roots are its coefficients?</a>, Sep 3 2015.
%H P. R. Stein, <a href="http://www.jstor.org/stable/2315341">On Polynomial Equations with Coefficients Equal to Their Roots</a>, The American Mathematical Monthly, Vol. 73, No. 3 (Mar., 1966), pp. 272-274.
%F Equals ((9-sqrt(57))^(1/3))/(3^(2/3)) + 2/((3(9-sqrt(57)))^(1/3)) (from Wolfram Alpha).
%F From _Wolfdieter Lang_, Sep 13 2022: (Start)
%F 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)].
%F Equals (1 + (1/9)*sqrt(57))^(1/3) + (1 - (1/9)*sqrt(57))^(1/3).
%F Equals (2/3)*sqrt(6)*cosh((1/3)*arccosh((3/4)*sqrt(6))). (End)
%e 1.7692923542386314152404094643350334926705530458988570042331061304026738...
%t RealDigits[Root[x^3-2x+2,1],10,120][[1]] (* _Harvey P. Dale_, May 25 2022 *)
%o (PARI) default(realprecision, 200);
%o -solve(x = -1.8, -1.7, x^3 - 2*x + 2)
%Y Cf. A273065, A273067, A357102.
%K nonn,cons
%O 1,2
%A _Rick L. Shepherd_, May 14 2016
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