%I #13 May 28 2023 13:43:53
%S 1,7,9,2,5,1,7,2,1,3,9,7,4,3,4,0,2,9,1,7,3,0,1,7,3,3,1,4,9,4,1,4,1,5,
%T 5,8,5,7,5,9,8,7,3,8,2,6,2,9,1,3,2,2,0,1,9,9,0,1,2,7,7,0,6,3,9,2,5,8,
%U 4,5,3,7,8,4,1,6,1,0,5,3,4,8,6,3,2,1
%N Decimal expansion of the least x such that 1/x + 1/(x+1) + 1/(x+2) = 3.
%C Equivalently, the least root of 3*x^3 + 6*x^2 - 2;
%C Middle root: A316247;
%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 *)
%t RealDigits[Root[1/x+1/(x+1)+1/(x+2)-3,1],10,120][[1]] (* _Harvey P. Dale_, May 28 2023 *)
%Y Cf. A305328, A316247, A316248.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Aug 19 2018
%E a(86) corrected by _Andrew Howroyd_, Nov 04 2018