|
%I #5 Jul 21 2018 23:37:09
%S 1,7,4,4,6,4,4,2,8,5,9,0,5,0,3,9,3,8,1,3,9,6,4,6,8,2,6,5,2,2,7,4,2,4,
%T 6,2,0,5,8,4,0,3,2,9,1,9,7,4,1,4,9,6,5,5,7,7,6,8,2,8,3,2,2,7,5,8,5,3,
%U 3,7,4,6,7,0,7,1,3,0,8,2,0,9,6,7,1,7
%N Decimal expansion of the least x such that 1/x + 1/(x+1) + 1/(x+2) = 2.
%C Equivalently, the least root of 2*x^3 + 3*x^2 - 2*x - 2;
%C Middle root: A316162;
%C Greatest root: A316163.
%C See A305328 for a guide to related sequences.
%F greatest root: -(1/2) + sqrt[7/3] cos[1/3 arctan[(2 sqrt[79/3])/3]]
%F middle: -(1/2) - 1/2 sqrt[7/3] cos[1/3 arctan[(2 sqrt[79/3])/3]] + 1/2 sqrt[7] sin[1/3 arctan[(2 sqrt[79/3])/3]]
%F least: -(1/2) - 1/2 sqrt[7/3] cos[1/3 arctan[(2 sqrt[79/3])/3]] - 1/2 sqrt[7] sin[1/3 arctan[(2 sqrt[79/3])/3]]
%e greatest root: 0.88922855912919436594...
%e middle root: -0.64458427322415498454...
%e least root: -1.7446442859050393814...
%t a = 1; b = 1; c = 1; u = 0; v = 1; w = 2; d = 2;
%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]]] (* A316161, least *)
%t RealDigits[u[[2]]] (* A316162, middle *)
%t RealDigits[u[[3]]] (* A316163, greatest *)
%Y Cf. A305328, A316162, A316163.
%K nonn,cons
%O 0,2
%A _Clark Kimberling_, Jul 21 2018
|
