A316247 - OEIS

%I #6 Aug 19 2018 18:20:06

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

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

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

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

%C Equivalently, the middle root of 3*x^3 + 6*x^2 - 2;

%C Least root: A316246;

%C Greatest root: A316248.

%C See A305328 for a guide to related sequences.

%F greatest root: -2/3 + (4/3)*cos((1/3)*arctan(3*sqrt(7)))

%F ****

%F middle: -2/3 - (2/3)*cos((1/3)*arctan(3*sqrt(7))) + (2*sin((1/3)*arctan(3*sqrt(7))))/sqrt(3)

%F ****

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

%e greatest root: 0.5148689384387165869...

%e middle root: -0.7223517244643762951...

%e least root: -1.792517213974340291...

%t a = 1; b = 1; c = 1; u = 0; v = 1; w = 2; 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 u = N[t, 200];

%t RealDigits[u[[1]]]  (* A316246, greatest *)

%t RealDigits[u[[2]]]  (* A316247, least *)

%t RealDigits[u[[3]]]  (* A316248, middle *)

%Y Cf. A305328, A316246, A316248.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Aug 19 2018