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%I #11 Sep 18 2018 02:21:23
%S 2,6,2,3,3,2,4,9,0,1,9,9,8,1,3,1,6,3,4,2,9,9,6,1,3,8,1,1,8,8,9,9,9,7,
%T 4,6,8,7,2,6,5,8,2,7,8,7,6,7,5,6,1,1,5,2,9,8,3,9,9,4,0,5,0,9,8,9,1,7,
%U 7,6,6,4,1,9,0,0,4,0,9,2,4,0,7,7,7,8
%N Decimal expansion of the middle x such that 1/x + 1/(x+3) + 1/(x+4) = 3 (negated).
%C Equivalently, the least root of 3*x^3 + 18*x^2 + 22 x - 12.
%C Least:: A316258
%C Greatest: A316260;
%C See A305328 for a guide to related sequences.
%H Muniru A Asiru, <a href="/A316259/b316259.txt">Table of n, a(n) for n = 1..1000</a>
%F greatest root: -2 + (2/3) sqrt(14) cos((1/3) arctan(sqrt(181/2)/9))
%F ****
%F middle: -2 - (1/3)sqrt(14) cos((1/3) arctan(sqrt(181/2)/9)) + sqrt(14/3) sin((1/3) arctan(sqrt(181/2)/9))
%F ****
%F least: -2 - 1/3 sqrt(14) cos((1/3) arctan(sqrt(181/2)/9)) - sqrt(14/3) sin((1/3) arctan(sqrt(181/2)/9))
%e greatest root: 0.4033761543003640184...
%e middle root: -2.623324901998131634... [Corrected by _A.H.M. Smeets_, Sep 17 2018]
%e least root: -3.780051252302232384...
%t a = 1; b = 1; c = 1; u = 0; v = 3; 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]]] (* A316259, middle *)
%t RealDigits[y[[2]]] (* A316258, least *)
%t RealDigits[y[[3]]] (* A316260, greatest *)
%Y Cf. A305328, A316258, A316260.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Sep 14 2018
%E Name corrected by _A.H.M. Smeets_, Sep 17 2018
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