%I #6 May 31 2018 21:36:51
%S 2,2,1,4,3,1,9,7,4,3,3,7,7,5,3,5,1,8,7,4,1,5,4,9,7,7,0,0,8,4,8,5,8,0,
%T 4,8,8,9,0,7,9,1,9,6,3,7,2,1,9,4,9,9,4,3,4,3,3,1,3,8,2,3,1,6,5,0,9,1,
%U 2,8,0,4,6,4,3,3,2,6,6,2,7,4,7,9,5,9
%N Decimal expansion of the greatest x such that 1/x + 1/(x+1) + 1/(x+2) = 1.
%C Equivalently, the greatest root of x^3 - 4*x - 2;
%C Middle root: A305327;
%C Least root: A305328.
%F greatest: (4 cos[1/3 arctan[sqrt[37/3]/3]])/sqrt[3]
%F middle:
%F -((2 cos[1/3 arctan[sqrt[37/3]/3]])/sqrt[3]) + 2 sin[1/3 arctan[sqrt[37/3]/3]]
%F least:
%F -((2 cos[1/3 arctan[sqrt[37/3]/3]])/sqrt[3]) - 2 sin[1/3 arctan[sqrt[37/3]/3]]
%e greatest root: 2.214319743377535187...
%e middle root: -0.539188872810889116...
%e least root: -1.67513087056664607088...
%t r[x_] := 1/x + 1/(x + 1) + 1/(x + 2);
%t -Numerator[Factor[r[x] - 1]]
%t t = x /. ComplexExpand[Solve[r[x] == 1, x]]
%t u = N[t, 120]
%t RealDigits[u[[1]]] (* A305326 *)
%t RealDigits[u[[2]]] (* A305327 *)
%t RealDigits[u[[3]]] (* A305328 *)
%Y Cf. A305327, A305328.
%K nonn,easy,cons
%O 1,1
%A _Clark Kimberling_, May 30 2018