A316138 - OEIS

%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