A358182 Decimal expansion of the real root of 2*x^3 - x^2 - x - 1.

1, 2, 3, 3, 7, 5, 1, 9, 2, 8, 5, 2, 8, 2, 5, 8, 7, 8, 8, 1, 9, 0, 9, 4, 3, 3, 7, 7, 6, 7, 9, 3, 9, 3, 0, 3, 5, 1, 9, 1, 1, 2, 7, 2, 3, 7, 5, 3, 1, 1, 8, 6, 4, 9, 4, 2, 3, 2, 0, 0, 9, 8, 8, 7, 0, 2, 7, 5, 3, 7, 5, 9, 6, 7, 9, 5

Offset: 1

Author

Wolfdieter Lang, Nov 07 2022

Keywords

• nonn
• cons
• easy

Comments

This equals r0 + 1/6 where r0 is the real root of y^3 - (7/12)*y - 16/27.

The other roots of 2*x^3 - x^2 - x - 1 are (1 + w1*(64 + 3*sqrt(417))^(1/3) + w2*(64 - 3*sqrt(417))^(1/3))/6 = -0.3668759642... + 0.5202594388...*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.

Using hyperbolic functions these roots are (1 - sqrt(7)*( cosh((1/3)*arccosh((64/49)*sqrt(7))) - sqrt(3)*sinh((1/3)*arccosh((64/49)*sqrt(7)))*i))/6, and its complex conjugate.

Examples

			1.23375192852825878819094337767939303519112723753118649423200988702753759...
		

Crossrefs

Cf. A358181, A358183.

Programs

Mathematica

RealDigits[x /. FindRoot[2*x^3 - x^2 - x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Nov 08 2022 *)

Formula

r = (1 + (64 + 3*sqrt(417))^(1/3) + 7*(64 + 3*sqrt(417))^(-1/3))/6.

r = (1 + (64 + 3*sqrt(417))^(1/3) + (64 - 3*sqrt(417))^(1/3))/6.

r = (1 + 2*sqrt(7)*cosh((1/3)*arccosh((64/49)*sqrt(7))))/6.

r = (1/6) + (4/3)*Hyper2F1([-1/6,1/3],[1/2],3753/4096). - Gerry Martens, Nov 08 2022