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%I #20 Oct 09 2023 12:03:11
%S 3,2,2,1,8,5,3,5,4,6,2,6,0,8,5,5,9,2,9,1,1,4,7,0,7,1,0,7,0,4,0,3,1,9,
%T 8,4,9,3,1,6,4,4,3,8,2,8,9,9,5,8,4,0,0,9,1,7,8,8,4,3,9,1,1,9,0,4,2,9,
%U 6,7,6,2,3,1,2,7,8,6
%N Decimal expansion of the real root of x^3 + 3*x - 1.
%C The other two roots are w1*phi^(1/3) - w2*(-1 + phi)^(1/3) = -0.16109267... + 1.75438095...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1; phi = A001622.
%C The hyperbolic function version is -sinh((1/3)*arcsinh(1/2)) + sqrt(3)*cosh((1/3)*arcsinh(1/2))*i, and its complex conjugate.
%F r = phi^(1/3) - phi^(-1/3), with phi = A001622.
%F r = phi^(1/3) - (-1 + phi)^(1/3).
%F r = 2*sinh((1/3)*arcsinh(1/2)).
%e 0.32218535462608559291147071070403198493164438289958400917884391190429676...
%p h := ((1 + sqrt(5))/2)^(1/3): evalf(h - 1/h, 90); # _Peter Luschny_, Sep 24 2022
%t RealDigits[Subtract @@ Surd[GoldenRatio, {3, -3}], 10, 100][[1]] (* _Amiram Eldar_, Sep 21 2022 *)
%t RealDigits[Root[x^3+3x-1,1],10,120][[1]] (* _Harvey P. Dale_, Oct 09 2023 *)
%o (PARI) 2*sinh((1/3)*asinh(1/2)) \\ _Michel Marcus_, Sep 23 2022
%Y Cf. A001622, A130880, A332437, A332438 - 2.
%K nonn,cons,easy
%O 0,1
%A _Wolfdieter Lang_, Sep 21 2022