A210462 - OEIS

%I #26 Oct 01 2024 20:13:06

%S 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,

%T 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,

%U 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

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

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

%C Real root of 8x^3 - 8x^2 + 2x - 1: an algebraic number of degree 3. - _Charles R Greathouse IV_, Apr 14 2014

%C The denominator of this algebraic number is 2, since its double is an algebraic integer. - _Charles R Greathouse IV_, Nov 12 2014

%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>

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

%e 0.87743883312334638002475444817926...

%p A075778neg := proc()

%p         1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;

%p end proc:

%p A210462 := proc()

%p         local a075778;

%p         a075778 := A075778neg() ;

%p         (1+1/a075778/(a075778-1))/2 ;

%p end proc:

%p evalf(A210462()) ;

%t (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 *)

%o (PARI) real(polroots(x^3-x^2+1))[2] \\ _Charles R Greathouse IV_, Apr 14 2014

%o (PARI) polrootsreal(8*x^3-8*x^2+2*x-1)[1] \\ _Charles R Greathouse IV_, Apr 14 2014

%Y Cf. A075778, A210463.

%K cons,nonn,easy,changed

%O 0,1

%A _R. J. Mathar_, Jan 22 2013