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A357469 Decimal expansion of the real root of x^3 - x^2 + x - 2.

%I A357469 #18 Aug 14 2023 10:33:44
%S A357469 1,3,5,3,2,0,9,9,6,4,1,9,9,3,2,4,4,2,9,4,8,3,1,0,1,3,3,2,5,7,7,3,8,8,
%T A357469 4,5,7,2,7,0,7,0,5,6,1,3,8,5,6,8,4,6,8,2,6,8,0,6,6,9,3,0,4,2,6,5,1,5,
%U A357469 1,8,9,7,2,3,2,2,0,9,2,0,8,5,9,1,6,5,8,0,3,9,7,7
%N A357469 Decimal expansion of the real root of x^3 - x^2 + x - 2.
%C A357469 This equals r0 + 1/3 where r0 is the real root of y^3 + (2/3)*y - 47/27, after 1/3.
%C A357469 The other (complex) roots of x^3 - x^2 + x - 2 are (w1*(4*(47 + 3*sqrt(249)))^(1/3) + (4*(47 - 3*sqrt(249)))^(1/3) + 2)/6 = -0.1766049820... + 1.2028208192...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.
%C A357469 Using hyperbolic functions these roots are (-sqrt(2)*(sinh((1/3)*arcsinh((47/8)*sqrt(2))) - sqrt(3)*cosh((1/3)*arcsinh((47/8)*sqrt(2)))*i) + 1)/3, and its complex conjugate.
%H A357469 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F A357469 r = ((4*(47 + 3*sqrt(249)))^(1/3) - 8*(4*(47 + 3*sqrt(249)))^(-1/3) + 2)/6.
%F A357469 r = ((4*(47 + 3*sqrt(249)))^(1/3) + w1*(4*(47 - 3*sqrt(249)))^(1/3) + 2)/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.
%F A357469 r = (2*sqrt(2)*sinh((1/3)*arcsinh((47/8)*sqrt(2))) + 1)/3.
%F A357469 Equals A197032 minus one.
%e A357469 1.3532099641993244294831013325773884572707056138568468268066930426515189723220920859165...
%p A357469 Digits := 140 ;
%p A357469 r := (2*sqrt(2)*sinh((1/3)*arcsinh((47/8)*sqrt(2))) + 1)/3 ;
%p A357469 evalf(%) ; # _R. J. Mathar_, Nov 08 2022
%t A357469 RealDigits[x /. FindRoot[x^3 - x^2 + x - 2, {x, 1}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Oct 18 2022 *)
%Y A357469 Cf. A197032, A137421, A357468.
%K A357469 nonn,cons,easy
%O A357469 1,2
%A A357469 _Wolfdieter Lang_, Oct 17 2022