A316167 - OEIS

%I #11 Aug 11 2018 11:30:54

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

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

%U 0,0,7,1,6,2,4,0,9,9,6,9,6,5,5,7,3,2

%N Decimal expansion of the greatest x such that 1/x + 1/(x+2) + 1/(x+4) = 2.

%C Equivalently, the least root of 2*x^3 + 9*x^2 + 4*x - 8;

%C Middle root: A316168;

%C Greatest root: A316169.

%C See A305328 for a guide to related sequences.

%F greatest root: -(3/2) + sqrt(19/3) cos(1/3 arctan((4 sqrt(427/3))/3))

%F middle root: -(3/2) - 1/2 sqrt(19/3) cos(1/3 arctan((4 sqrt(427/3))/3)) + 1/2 sqrt(19) sin(1/3 arctan((4 sqrt(427/3))/3))

%F least root: -(3/2) - 1/2 sqrt(19/3) cos(1/3 arctan((4 sqrt(427/3))/3)) - 1/2 sqrt(19) sin(1/3 arctan((4 sqrt(427/3))/3))

%e greatest root: 0.70530340009105630377...

%e middle root: -1.5526623262135260618...

%e least root: -3.6526410738775302420...

%t a = 1; b = 1; c = 1; u = 0; v = 2; w = 4; 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]]]  (* A316167, greatest *)

%t RealDigits[u[[2]]]  (* A316168, middle *)

%t RealDigits[u[[3]]]  (* A316169, least *)

%o (PARI) solve(x=0, 1, 2*x^3 + 9*x^2 + 4*x - 8) \\ _Michel Marcus_, Aug 11 2018

%Y Cf. A305328, A316168, A316169.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Aug 09 2018