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A358183 Decimal expansion of the real root of 2*x^3 + x^2 - x - 1.

%I A358183 #20 Jun 08 2025 19:54:58
%S A358183 8,2,9,4,8,3,5,4,0,9,5,8,4,9,7,0,3,9,6,7,3,3,8,7,5,7,8,3,9,2,0,0,7,8,
%T A358183 0,7,6,2,1,9,9,6,6,7,2,2,8,1,3,8,8,5,5,0,1,7,6,1,0,7,7,4,4,4,9,2,0,8,
%U A358183 4,0,1,0,3,9,0,1
%N A358183 Decimal expansion of the real root of 2*x^3 + x^2 - x - 1.
%C A358183 This equals r0 - 1/6 where r0 is the real root of y^3 - (7/12)*y - 11/27.
%C A358183 The other (complex) roots of 2*x^3 + x^2 - x - 1 are (-1 + w1*(44 + 3*sqrt(177))^(1/3) + w2*(44 - 3*sqrt(177))^(1/3))/6 = -0.6647417704... + 0.4011272787...*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 conjugate roots of x^3 - 1.
%C A358183 Using hyperbolic functions these roots are  (-1 - sqrt(7)*(cosh((1/3)*arccosh((44/49)*sqrt(7))) - sqrt(3)*sinh((1/3)*arccosh((44/49)*sqrt(7)))*i))/6, and its complex conjugate.
%H A358183 Harvey P. Dale, <a href="/A358183/b358183.txt">Table of n, a(n) for n = 0..1000</a>
%F A358183 r = (-1 + (44 + 3*sqrt(177))^(1/3) + 7*(44 + 3*sqrt(177))^(-1/3))/6.
%F A358183 r = (-1 + (44 + 3*sqrt(177))^(1/3) + (44 - 3*sqrt(177))^(1/3))/6.
%F A358183 r = (-1 + 2*sqrt(7)*cosh((1/3)*arccosh((44/49)*sqrt(7))))/6.
%F A358183 r = (-1/6) + (2^(2/3)*11^(1/3))/3 * Hyper2F1([-1/6,1/3],[1/2],1593/1936). - _Gerry Martens_, Nov 08 2022
%e A358183 0.82948354095849703967338757839200780762199667228138855017610774449208401039...
%t A358183 RealDigits[x /. FindRoot[2*x^3 + x^2 - x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Nov 08 2022 *)
%t A358183 RealDigits[Root[2x^3+x^2-x-1,1],10,120][[1]] (* _Harvey P. Dale_, Jun 08 2025 *)
%o A358183 (PARI) (-1/6) + (2^(2/3)*11^(1/3))/3 * hypergeom([-1/6,1/3],[1/2],1593/1936) \\ _Michel Marcus_, Nov 08 2022
%Y A358183 Cf. A358182, A358184.
%K A358183 nonn,cons,easy
%O A358183 0,1
%A A358183 _Wolfdieter Lang_, Nov 07 2022