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A357101 Decimal expansion of the real root of x^3 - 2*x^2 - 2.

%I A357101 #24 Mar 23 2025 20:51:38
%S A357101 2,3,5,9,3,0,4,0,8,5,9,7,1,7,7,6,4,2,0,7,3,0,6,6,0,3,9,2,8,0,0,5,3,2,
%T A357101 5,5,5,3,4,6,4,8,1,2,7,8,0,6,7,6,7,2,2,8,3,7,9,7,1,4,1,2,5,1,5,8,3,8,
%U A357101 7,5,5,8,8,9,4,4,6,5
%N A357101 Decimal expansion of the real root of x^3 - 2*x^2 - 2.
%C A357101 This equals r0 + 2/3 where r0 is the real root of y^3 - (4/3)*y - 70/27.
%C A357101 The other two roots are (w1*(35 + 3*sqrt(129))^(1/3) + w2*(35 - 3*sqrt(129))^(1/3) + 2)/3 = -0.1796520429... + 0.9030131458...*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.
%C A357101 Using hyperbolic functions these roots are (2/3)*(1 - cosh((1/3)*arccosh(35/8)) + sqrt(3)*sinh((1/3)*arccosh(35/8))*i), and its complex conjugate.
%H A357101 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F A357101 r = ((35 + 3*sqrt(129))^(1/3) + 4*(35 + 3*sqrt(129))^(-1/3) + 2)/3.
%F A357101 r = ((35 + 3*sqrt(129))^(1/3) + (35 - 3*sqrt(129))^(1/3) + 2)/3.
%F A357101 r = (2/3)*(2*cosh((1/3)*arccosh(35/8)) + 1).
%e A357101 2.359304085971776420730660392800532555346481278067672283797141251583875588...
%p A357101 h := ((35 + 3*sqrt(129))/8)^(1/3): evalf((1 + h + 1/h)*2/3, 82); # _Peter Luschny_, Sep 25 2022
%t A357101 RealDigits[x /. FindRoot[x^3 - 2*x^2 - 2, {x, 2}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Sep 21 2022 *)
%o A357101 (PARI) polrootsreal(x^3 - 2*x^2 - 2)[1] \\ _Michel Marcus_, Sep 23 2022
%Y A357101 Cf. A058265 - 1 (x^3 + 2*x^2 - 2).
%K A357101 nonn,cons,easy
%O A357101 1,1
%A A357101 _Wolfdieter Lang_, Sep 20 2022