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

%I A210462 #29 Mar 25 2025 15:29:28
%S A210462 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 A210462 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 A210462 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 A210462 Decimal expansion of the real part of the complex roots of x^3-x^2+1.
%C A210462 The real root is A075778 (negated). The imaginary parts are plus or minus A210463.
%C A210462 Real root of 8x^3 - 8x^2 + 2x - 1: an algebraic number of degree 3. - _Charles R Greathouse IV_, Apr 14 2014
%C A210462 The denominator of this algebraic number is 2, since its double is an algebraic integer. - _Charles R Greathouse IV_, Nov 12 2014
%H A210462 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F A210462 Equals 1/2 + 1/(2*A075778*(A075778+1)).
%e A210462 0.87743883312334638002475444817926...
%p A210462 A075778neg := proc()
%p A210462         1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;
%p A210462 end proc:
%p A210462 A210462 := proc()
%p A210462         local a075778;
%p A210462         a075778 := A075778neg() ;
%p A210462         (1+1/a075778/(a075778-1))/2 ;
%p A210462 end proc:
%p A210462 evalf(A210462()) ;
%t A210462 (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 A210462 (PARI) real(polroots(x^3-x^2+1))[2] \\ _Charles R Greathouse IV_, Apr 14 2014
%o A210462 (PARI) polrootsreal(8*x^3-8*x^2+2*x-1)[1] \\ _Charles R Greathouse IV_, Apr 14 2014
%Y A210462 Cf. A075778, A210463.
%K A210462 cons,nonn,easy
%O A210462 0,1
%A A210462 _R. J. Mathar_, Jan 22 2013