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Decimal expansion of the greatest x such that 1/x + 1/(x+1) + 1/(x+2) = 2.
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%I #7 Sep 13 2018 08:03:45

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

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

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

%N Decimal expansion of the greatest x such that 1/x + 1/(x+1) + 1/(x+2) = 2.

%C Equivalently, the least root of 2*x^3 + 3*x^2 - 2*x - 2;

%C Middle root: A316162;

%C Greatest root: A316163.

%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: -(1/2) - 1/2 sqrt(7/3) cos(1/3 arctan((2 sqrt(79/3))/3)) + 1/2 sqrt(7) sin(1/3 arctan((2 sqrt(79/3))/3))

%F least: -(1/2) - 1/2 sqrt(7/3) cos(1/3 arctan((2 sqrt(79/3))/3)) - 1/2 sqrt(7) sin(1/3 arctan((2 sqrt(79/3))/3))

%e greatest root: 0.88922855912919436594...

%e middle root: -0.64458427322415498454...

%e least root: -1.7446442859050393814...

%t a = 1; b = 1; c = 1; u = 0; v = 1; w = 2; d = 2;

%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]]] (* A316161, least *)

%t RealDigits[u[[2]]] (* A316162, middle *)

%t RealDigits[u[[3]]] (* A316163, greatest *)

%Y Cf. A305328, A316161, A316162.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Aug 08 2018