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Decimal expansion of the middle x such that 1/x + 1/(x+2) + 1/(x+4) = 3.
4

%I #5 Sep 16 2018 21:44:26

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

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

%U 2,1,5,3,6,3,3,6,9,2,6,8,6,1,0,2,3,9

%N Decimal expansion of the middle x such that 1/x + 1/(x+2) + 1/(x+4) = 3.

%C Equivalently, the least root of 3*x^3 + 15*x^2 + 12 x - 8.

%C Least: A316255;

%C Greatest: A316257.

%C See A305328 for a guide to related sequences.

%F greatest root: -(5/3) + (2/3) sqrt(13) cos((1/3) arctan(6 sqrt(61)))

%F ****

%F middle: -(5/3) - (1/3) sqrt(13) cos((1/3) arctan(6 sqrt(61))) - sqrt(13/3) sin((1/3) arctan(6 sqrt(61)))

%F ****

%F least: -(5/3) - (1/3) sqrt(13) cos((1/3) arctan(6 sqrt(61))) + sqrt(13/3) sin((1/3) arctan(6 sqrt(61)))

%e greatest root: 0.4234942709347976489...

%e middle root: -1.683761836678034312...

%e least root: -3.739732434256763336...

%t a = 1; b = 1; c = 1; u = 0; v = 2; w = 4; d = 3;

%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 y = Re[N[t, 200]];

%t RealDigits[y[[1]]] (* A316257, greatest *)

%t RealDigits[y[[2]]] (* A316255, least *)

%t RealDigits[y[[3]]] (* A316256, middle *)

%Y Cf. A305328, A316255, A316257.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Sep 14 2018