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A273065 Decimal expansion of the negative reciprocal of the real root of x^3 - 2x + 2.

%I A273065 #19 Feb 11 2025 09:26:53
%S A273065 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,
%T A273065 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,
%U A273065 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
%N A273065 Decimal expansion of the negative reciprocal of the real root of x^3 - 2x + 2.
%C A273065 The roots of x^3 + A273065*x^2 - A273066*x + A273067 are A273065, -A273066, and A273067. See A273066, the main entry.
%C A273065 From _Wolfdieter Lang_, Sep 15 2022: (Start)
%C A273065 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.
%C A273065 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.
%C A273065 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)
%H A273065 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>.
%F A273065 Equals 1/A273066.
%F A273065 From _Wolfdieter Lang_, Sep 15 2022: (Start)
%F A273065 Equals ((23/4 + (3/4)*sqrt(57))^(1/3) + (23/4 + (3/4)*sqrt(57))^(-1/3) - 1)/3.
%F A273065 Equals ((23/4 + (3/4)*sqrt(57))^(1/3) + (23/4 - (3/4)*sqrt(57))^(1/3) - 1)/3.
%F A273065 Equals (2*cosh((1/3)*arccosh(23/4)) - 1)/3. (End)
%e A273065 0.565197717383639396437528013247030816098483976759553827555483810948411203...
%t A273065 First[RealDigits[1/x/.N[First[Solve[x^3-2x+2==0,x]],105]]] (* _Stefano Spezia_, Sep 15 2022 *)
%o A273065 (PARI) default(realprecision, 200);
%o A273065 -1/solve(x = -1.8, -1.7, x^3 - 2*x + 2)
%Y A273065 Cf. A273066, A273067, A357109.
%K A273065 nonn,cons
%O A273065 0,1
%A A273065 _Rick L. Shepherd_, May 15 2016