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A210463 Decimal expansion of the absolute value of the imaginary part of the two complex roots of x^3-x^2+1.

%I A210463 #22 Mar 25 2025 00:57:43
%S A210463 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 A210463 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 A210463 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 A210463 Decimal expansion of the absolute value of the imaginary part of the two complex roots of x^3-x^2+1.
%C A210463 An algebraic number of degree 6. - _Charles R Greathouse IV_, Apr 14 2014
%C A210463 The denominator of this algebraic number is 2, since its double is an algebraic integer. - _Charles R Greathouse IV_, Nov 12 2014
%H A210463 <a href="/index/Al#algebraic_06">Index entries for algebraic numbers, degree 6</a>
%F A210463 Equals sqrt(1/A075778-A210462^2).
%e A210463 0.744861766619744236593170428604392367240163...
%p A210463 A075778neg := proc()
%p A210463         1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;
%p A210463 end proc:
%p A210463 A210463 := proc()
%p A210463         local a075778,a210462 ;
%p A210463         a075778 := A075778neg() ;
%p A210463         a210462 := A210462() ;
%p A210463         -1/a075778-a210462^2 ;
%p A210463         sqrt(%) ;
%p A210463 end proc:
%p A210463 evalf(A210463()) ;
%t A210463 -((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 A210463 (PARI) polrootsreal(64*x^6+32*x^4+4*x^2-23)[2] \\ _Charles R Greathouse IV_, Apr 14 2014
%Y A210463 Cf. A075778, A210462.
%K A210463 cons,easy,nonn
%O A210463 0,1
%A A210463 _R. J. Mathar_, Jan 22 2013