A210462 Decimal expansion of the real part of the complex roots of x^3-x^2+1.

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

Offset: 0

Author

R. J. Mathar, Jan 22 2013

Keywords

• cons
• nonn
• easy

Comments

The real root is A075778 (negated). The imaginary parts are plus or minus A210463.

Real root of 8x^3 - 8x^2 + 2x - 1: an algebraic number of degree 3. - Charles R Greathouse IV, Apr 14 2014

The denominator of this algebraic number is 2, since its double is an algebraic integer. - Charles R Greathouse IV, Nov 12 2014

Examples

			0.87743883312334638002475444817926...
		

Links

• Index entries for algebraic numbers, degree 3

Crossrefs

Cf. A075778, A210463.

Programs

Maple

A075778neg := proc()
        1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;
end proc:
A210462 := proc()
        local a075778;
        a075778 := A075778neg() ;
        (1+1/a075778/(a075778-1))/2 ;
end proc:
evalf(A210462()) ;

Mathematica

(2^(2/3)*(25 + 3*Sqrt[69])^(1/3) + 2^(2/3)*(25 - 3*Sqrt[69])^(1/3) + 4)/12 // RealDigits[#, 10, 125]& // First (* Jean-François Alcover, Feb 20 2013 *)

PARI

real(polroots(x^3-x^2+1))[2] \\ Charles R Greathouse IV, Apr 14 2014

PARI

polrootsreal(8*x^3-8*x^2+2*x-1)[1] \\ Charles R Greathouse IV, Apr 14 2014

Formula

Equals 1/2 + 1/(2*A075778*(A075778+1)).