A316167 Decimal expansion of the greatest x such that 1/x + 1/(x+2) + 1/(x+4) = 2.

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

Offset

0,1

Comments

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

Middle root: A316168;

Greatest root: A316169.

See A305328 for a guide to related sequences.

Links

Table of n, a(n) for n=0..85.

Formula

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

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))

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))

Example

greatest root: 0.70530340009105630377...
middle root: -1.5526623262135260618...
least root: -3.6526410738775302420...

Mathematica

a = 1; b = 1; c = 1; u = 0; v = 2; w = 4; d = 2;
r[x_] := a/(x + u) + b/(x + v) + c/(x + w);
t = x /. ComplexExpand[Solve[r[x] == d, x]]
N[t, 20]
u = N[t, 200];
RealDigits[u[[1]]]  (* A316167, greatest *)
RealDigits[u[[2]]]  (* A316168, middle *)
RealDigits[u[[3]]]  (* A316169, least *)

Prog

(PARI) solve(x=0, 1, 2*x^3 + 9*x^2 + 4*x - 8) \\ Michel Marcus, Aug 11 2018

Crossrefs

Cf. A305328, A316168, A316169.

Sequence in context: A064648 A116198 A137915 * A272037 A309724 A198114

Adjacent sequences: A316164 A316165 A316166 * A316168 A316169 A316170

Keyword

nonn,cons

Author

Clark Kimberling, Aug 09 2018

Status

approved