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%I #10 Sep 13 2018 08:04:23
%S 2,6,6,4,0,1,5,6,4,4,7,4,7,8,0,4,5,9,2,5,4,6,7,4,8,4,2,8,0,4,5,4,4,0,
%T 2,2,2,1,4,3,9,4,1,8,1,8,3,5,4,5,6,7,0,1,2,3,2,2,2,1,8,0,4,4,7,8,3,6,
%U 5,8,4,9,9,5,1,1,2,1,6,0,6,6,5,0,0,5
%N Decimal expansion of the least x such that 1/x + 1/(x+1) + 1/(x+3) = 2.
%C Equivalently, the least root of 2*x^3 + 5*x^2 - 2*x - 3;
%C Middle root: A316165;
%C Greatest root: A316166.
%C See A305328 for a guide to related sequences.
%F greatest root: -(5/6) + (1/6) sqrt(37) cos((1/3)(Pi - arctan((6 sqrt(1329))/53))) + (1/6) sqrt(37) cos((1/3)(-Pi + arctan((6 sqrt(1329))/53)))
%F ****
%F middle: -(5/6) - (1/12) sqrt(37) cos((1/3)(Pi - arctan((6 sqrt(1329))/53))) -
%F (1/12) sqrt(37) cos((1/3)(-Pi + arctan((6 sqrt(1329))/53))) +
%F (1/4) sqrt(37/3) sin((1/3)(Pi - arctan((6 sqrt(1329))/53))) -
%F (1/4) sqrt(37/3) sin((1/3)(-Pi + arctan((6 sqrt(1329))/53)))
%F ****
%F least: -(5/6) - (1/12) sqrt(37) cos((1/3)(Pi - arctan((6 sqrt(1329))/53))) -
%F (1/12) sqrt(37) cos(1/3(-Pi + arctan((6 sqrt(1329))/53))) -
%F (1/4) sqrt(37/3) sin((1/3)(Pi - arctan((6 sqrt(1329))/53))) +
%F (1/4) sqrt(37/3) sin(1/3)(-Pi + arctan((6 sqrt(1329))/53)))
%e greatest root: 0.83684889130097120054...
%e middle root: -0.67283324655316660799...
%e least root: -2.6640156447478045925...
%t a = 1; b = 1; c = 1; u = 0; v = 1; w = 3; d = 2;
%t r[x_] := a/(x + u) + b/(x + v) + c/(x + w);
%t t = Re[x /. ComplexExpand[Solve[r[x] == d, x]]]
%t N[t, 20]
%t u = N[t, 200];
%t u1 = RealDigits[u[[1]]] (* A316166, greatest *)
%t u2 = RealDigits[u[[2]]] (* A316164, least *)
%t u3 = RealDigits[u[[3]]] (* A316165, middle *)
%Y Cf. A305328, A316165, A316166.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Aug 08 2018
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