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%I #4 Jul 21 2018 23:36:57
%S 1,2,8,9,1,6,8,5,4,6,4,4,8,3,0,9,9,6,9,0,8,2,6,7,7,4,5,8,1,6,8,5,6,7,
%T 3,8,8,1,4,2,9,0,2,2,0,2,8,4,2,7,3,8,3,4,3,7,2,9,4,7,0,0,6,3,0,1,3,5,
%U 6,4,6,4,8,4,0,4,3,7,4,4,7,4,1,8,4,5
%N Decimal expansion of the middle x such that 1/x + 1/(x+2) + 1/(x+4) = 1.
%C Equivalently, the middle root of x^3 + 3*x^2 - 4*x - 8;
%C Least root: A316137
%C Middle root: A316138;
%C Greatest root: A316139.
%C See A305328 for a guide to related sequences.
%F greatest root: -1 + 2 sqrt[7/3] cos[1/3 arctan[(2 sqrt[79/3])/3]]
%F middle: -1 - sqrt[7/3] cos[1/3 arctan[(2 sqrt[79/3])/3]] + sqrt[7] sin[1/3 arctan[(2 sqrt[79/3])/3]]
%F least: -1 - sqrt[7/3] cos[1/3 arctan[(2 sqrt[79/3])/3]] - sqrt[7] sin[1/3 arctan[(2 sqrt[79/3])/3]]
%e greatest root: 1.7784571182583887319...
%e middle root: -1.2891685464483099691...
%e least root: -3.4892885718100787628...
%t a = 1; b = 1; c = 1; u = 0; v = 2; w = 4; d = 1;
%t r[x_] := a/(x + u) + b/(x + v) + c/(x + w);
%t t = x /. ComplexExpand[Solve[r[x] == d, x]]
%t N[t, 20]
%t u = N[t, 200];
%t RealDigits[u[[1]]] (* A316137, least *)
%t RealDigits[u[[2]]] (* A316138, middle *)
%t RealDigits[u[[3]]] (* A316139, greatest *)
%Y Cf. A305328, A316137, A316139.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Jul 21 2018
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