|
%I #6 Aug 19 2018 18:20:14
%S 5,1,4,8,6,8,9,3,8,4,3,8,7,1,6,5,8,6,8,9,5,6,7,5,4,6,4,1,9,7,8,6,1,2,
%T 5,0,0,4,7,6,6,8,7,2,9,8,8,1,3,5,0,3,4,8,8,1,5,8,1,6,3,3,7,6,1,3,8,7,
%U 5,1,6,7,5,9,7,2,3,1,3,4,2,4,7,8,1,2
%N Decimal expansion of the greatest x such that 1/x + 1/(x+1) + 1/(x+2) = 3.
%C Equivalently, the greatest root of 3*x^3 + 6*x^2 - 2;
%C Middle root: A316247;
%C Least root: A316246.
%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, A316247.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Aug 19 2018
|
