A316136 - OEIS

%I #6 Jul 21 2018 23:36:42

%S 1,8,6,6,1,9,8,2,6,2,5,0,9,0,2,2,5,0,5,4,7,3,9,9,4,2,3,0,9,8,6,5,9,1,

%T 4,0,6,2,3,5,9,0,0,7,5,5,7,7,2,8,8,0,4,6,1,9,7,8,4,1,6,7,0,4,8,6,1,1,

%U 1,7,0,7,6,1,0,8,7,1,6,1,9,7,8,2,6,3,9,5,0,3,0,6,6,1,2,8,2,2,5,0,2,8,0,0,0

%N Decimal expansion of the greatest x such that 1/x + 1/(x+2) + 1/(x+3) = 1.

%C Equivalently, the greatest root of x^3 + 2*x^2 - 4*x - 6;

%C Middle root: A316135;

%C Greatest root: A316136.

%C See A305328 for a guide to related sequences.

%F greatest root: -(2/3) + 8/3 cos[1/3 arctan[(3 sqrt[303])/37]]

%F middle: -(2/3) - 4/3 cos[1/3 arctan[(3 sqrt[303])/37]] + (4 sin[1/3 arctan[(3 sqrt[303])/37]])/sqrt[3]

%F least: -(2/3) - 4/3 cos[1/3 arctan[(3 sqrt[303])/37]] - (4 sin[1/3 arctan[(3 sqrt[303])/37]])/sqrt[3]

%e greatest root: 1.8661982625090225055...

%e middle root: -1.2107558809591917224...

%e least root: -2.6554423815498307831...

%t a = 1; b = 1; c = 1; u = 0; v = 2; w = 3; d = 1;

%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]]]  (* A316134, least *)

%t RealDigits[u[[2]]]  (* A316135, middle *)

%t RealDigits[u[[3]]]  (* A316136, greatest *)

%t RealDigits[8 Cos[ArcTan[3 Sqrt[303]/37]/3]/3 - 2/3, 10, 111][[1]] (* _Robert G. Wilson v_, Jul 21 2018 *)

%Y Cf. A305328, A316134, A316135.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Jul 21 2018