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%I #5 Sep 08 2018 22:14:47
%S 2,7,8,4,2,1,8,6,9,1,4,5,7,7,1,0,3,6,2,9,3,4,7,1,2,0,7,9,4,9,9,5,2,7,
%T 6,9,9,7,2,1,3,3,0,0,6,8,6,0,9,4,7,7,6,7,5,2,0,9,6,7,9,6,7,8,0,8,9,4,
%U 7,0,4,6,2,8,4,7,5,0,2,0,0,0,9,4,2,3
%N Decimal expansion of the least x such that 1/x + 1/(x+2) + 1/(x+3) = 3.
%C Equivalently, the least root of 3*x^3 + 12*x^2 + 8 x - 6;
%C Middle root: A316253;
%C Greatest root: A316254.
%C See A305328 for a guide to related sequences.
%F greatest root: -(4/3) + (4/3) sqrt(2) cos((1/3) arctan(sqrt(391)/11))
%F ****
%F middle: -(4/3) - (2/3) sqrt(2) cos((1/3) arctan(sqrt(391)/11)) + 2 sqrt(2/3) sin((1/3) arctan(sqrt(391)/11))
%F ****
%F least: -(4/3) - (2/3) sqrt(2) cos((1/3) arctan(sqrt(391)/11)) - 2 sqrt(2/3) sin((1/3) arctan(sqrt(391)/11))
%e greatest root: 0.4351172195495135109...
%e middle root: -1.650898528091803148...
%e least root: -2.784218691457710362...
%t a = 1; b = 1; c = 1; u = 0; v = 2; w = 3; 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]]] (* A316254, greatest *)
%t RealDigits[y[[2]]] (* A316252, least *)
%t RealDigits[y[[3]]] (* A316253, middle *)
%Y Cf. A305328, A316253, A316254.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Sep 08 2018
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