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

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, 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, 4, 0, 1, 0, 3, 9, 0, 1

Offset

0,1

Comments

This equals r0 - 1/6 where r0 is the real root of y^3 - (7/12)*y - 11/27.

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.

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.

Links

Table of n, a(n) for n=0..75.

Formula

r = (-1 + (44 + 3*sqrt(177))^(1/3) + 7*(44 + 3*sqrt(177))^(-1/3))/6.

r = (-1 + (44 + 3*sqrt(177))^(1/3) + (44 - 3*sqrt(177))^(1/3))/6.

r = (-1 + 2*sqrt(7)*cosh((1/3)*arccosh((44/49)*sqrt(7))))/6.

r = (-1/6) + (2^(2/3)*11^(1/3))/3 * Hyper2F1([-1/6,1/3],[1/2],1593/1936). - Gerry Martens, Nov 08 2022

Example

0.82948354095849703967338757839200780762199667228138855017610774449208401039...

Mathematica

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

Prog

(PARI) (-1/6) + (2^(2/3)*11^(1/3))/3 * hypergeom([-1/6, 1/3], [1/2], 1593/1936) \\ Michel Marcus, Nov 08 2022

Crossrefs

Cf. A358182, A358184.

Sequence in context: A181164 A154212 A155035 * A154189 A359086 A154459

Adjacent sequences: A358180 A358181 A358182 * A358184 A358185 A358186

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Nov 07 2022

Status

approved