A316168 Decimal expansion of the middle x such that 1/x + 1/(x+2) + 1/(x+4) = 2, negated.

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

Offset

1,2

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=1..86.

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

Crossrefs

Cf. A305328, A316167, A316169.

Sequence in context: A021185 A132376 A273007 * A019602 A153839 A322159

Adjacent sequences: A316165 A316166 A316167 * A316169 A316170 A316171

Keyword

nonn,cons

Author

Clark Kimberling, Aug 09 2018

Status

approved