A358182 -id:A358182 - cp's OEIS frontend

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

Wolfdieter Lang, Nov 07 2022

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.

			0.82948354095849703967338757839200780762199667228138855017610774449208401039...

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A358182, A358184.

RealDigits[x /. FindRoot[2*x^3 + x^2 - x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Nov 08 2022 *)
RealDigits[Root[2x^3+x^2-x-1,1],10,120][[1]] (* Harvey P. Dale, Jun 08 2025 *)

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

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

A358184 Decimal expansion of the real root of 2*x^3 - x^2 + x - 1.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Nov 07 2022

This equals r0 - 1/6 where r0 is the real root of y^3 - (7/12)*y - 11.
The other (complex) roots of 2*x^3 - x^2 + x - 1 are (1 + w1*(46 + 3*sqrt(249))^(1/3)  + (46 - 3*sqrt(249))^(1/3))/6 = -0.1194918107... + 0.8138345589...*i, and its conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex conjugate roots of x^3 - 1.
Using hyperbolic functions these roots are (1 - sqrt(5)*(sinh((1/3)*arcsinh((46/25)*sqrt(5))) - sqrt(3)*cosh((1/3)*arcsinh((46/25)*sqrt(5)))*i))/6, and its complex conjugate.

			0.73898362150450623732346254067108755072377477637909672211795496930230203...

Cf. A358182, A358183.

RealDigits[x /. FindRoot[2*x^3 - x^2 + x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Nov 08 2022 *)
RealDigits[Root[-1+x-x^2+2 x^3,1],10,120][[1]] (* Harvey P. Dale, Sep 09 2023 *)

r = (1 + (46 + 3*sqrt(249))^(1/3) - 5*(46+3*sqrt(249))^(-1/3))/6.
r = (1+ (46 + 3*sqrt(249))^(1/3)  + w1*(46 - 3*sqrt(249))^(1/3))/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3).
r = (1 + 2*sqrt(5)*sinh((1/3)*arcsinh((46/25)*sqrt(5))))/6.
r = (1/6) + (46/45)*Hyper2F1([1/3, 2/3],[3/2], -(46^2/5^3)). - Gerry Martens, Nov 08 2022

cp's OEIS Frontend

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

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