%I #19 Aug 21 2023 11:40:11

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

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

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

%N Decimal expansion of the absolute value of the imaginary part of the two complex roots of x^3-x^2+1.

%C An algebraic number of degree 6. - _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_06">Index entries for algebraic numbers, degree 6</a>

%F Equals sqrt(1/A075778-A210462^2).

%e 0.744861766619744236593170428604392367240163...

%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 A210463 := proc()

%p local a075778,a210462 ;

%p a075778 := A075778neg() ;

%p a210462 := A210462() ;

%p -1/a075778-a210462^2 ;

%p sqrt(%) ;

%p end proc:

%p evalf(A210463()) ;

%t -((2^(1/3)*(25 - 3*Sqrt[69])^(2/3) - 2)/(2*2^(2/3)*Sqrt[3]*(25 - 3*Sqrt[69])^(1/3))) // RealDigits[#, 10, 125]& // First (* _Jean-François Alcover_, Feb 20 2013 *)

%o (PARI) polrootsreal(64*x^6+32*x^4+4*x^2-23)[2] \\ _Charles R Greathouse IV_, Apr 14 2014

%Y Cf. A075778, A210462.

%K cons,easy,nonn

%O 0,1

%A _R. J. Mathar_, Jan 22 2013