A357100 Decimal expansion of the real root of x^3 + x^2 - 3.

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

Offset: 1

Author

Wolfdieter Lang, Sep 13 2022

Keywords

• nonn
• cons
• easy

Comments

This equals r0 - 1/3 where r0 is the real root of y^3 - (1/3)*y - 79/27.

The other two roots are (w1*(79/2 + (9/2)*sqrt(77))^(1/3) + w2*(79/2 - (9/2)*sqrt(77))^(1/3) - 1)/3 = -1.08727970... + 1.1713121...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.

Using hyperbolic function these roots are (-(1 + cosh((1/3)*arccosh(79/2))) + sqrt(3)*sinh((1/3)*arccosh(79/2))*i)/3, and its complex conjugate.

Examples

			r = 1.17455941029298007420231898869565392567594872533708249833673392030236...
		

Crossrefs

Cf. A356034.

Programs

Maple

Digits := 120: a := ((79 + 9*sqrt(77))/2)^(1/3): (a + 1/a - 1)/3: evalf(%)*10^96: ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Sep 15 2022

Mathematica

First[RealDigits[x/.N[First[Solve[x^3+x^2-3==0, x]], 76]]] (* Stefano Spezia, Sep 15 2022 *)

PARI

(2*cosh((1/3)*acosh(79/2)) - 1)/3 \\ Michel Marcus, Sep 15 2022

Formula

r = ((316 + 36*sqrt(77))^(1/3) + 4/(316 + 36*sqrt(77))^(1/3) - 2)/6.

r = ((79/2 + (9/2)*sqrt(77))^(1/3) + (79/2 - (9/2)*sqrt(77))^(1/3) - 1)/3.

r = (2*cosh((1/3)*arccosh(79/2)) - 1)/3.