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Decimal expansion of the middle x such that 1/x + 1/(x+1) + 1/(x+2) = 2.
4

%I #4 Jul 21 2018 23:37:15

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

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

%U 2,3,2,4,2,0,2,1,8,7,2,3,7,0,9,2,2,8

%N Decimal expansion of the middle 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, A316163.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Jul 21 2018