Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #17 Sep 29 2022 14:20:13
%S 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 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 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 Decimal expansion of the negative reciprocal of the real root of x^3 - 2x + 2.
%C The roots of x^3 + A273065*x^2 - A273066*x + A273067 are A273065, -A273066, and A273067. See A273066, the main entry.
%C From _Wolfdieter Lang_, Sep 15 2022: (Start)
%C 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 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 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)
%F Equals 1/A273066.
%F From _Wolfdieter Lang_, Sep 15 2022: (Start)
%F Equals ((23/4 + (3/4)*sqrt(57))^(1/3) + (23/4 + (3/4)*sqrt(57))^(-1/3) - 1)/3.
%F Equals ((23/4 + (3/4)*sqrt(57))^(1/3) + (23/4 - (3/4)*sqrt(57))^(1/3) - 1)/3.
%F Equals (2*cosh((1/3)*arccosh(23/4)) - 1)/3. (End)
%e 0.565197717383639396437528013247030816098483976759553827555483810948411203...
%t First[RealDigits[1/x/.N[First[Solve[x^3-2x+2==0,x]],105]]] (* _Stefano Spezia_, Sep 15 2022 *)
%o (PARI) default(realprecision, 200);
%o -1/solve(x = -1.8, -1.7, x^3 - 2*x + 2)
%Y Cf. A273066, A273067, A357109.
%K nonn,cons
%O 0,1
%A _Rick L. Shepherd_, May 15 2016