A316138 Decimal expansion of the middle x such that 1/x + 1/(x+2) + 1/(x+4) = 1.

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

Offset

1,2

Comments

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

Least root: A316137

Middle root: A316138;

Greatest root: A316139.

See A305328 for a guide to related sequences.

Links

Table of n, a(n) for n=1..86.

Formula

greatest root: -1 + 2 sqrt[7/3] cos[1/3 arctan[(2 sqrt[79/3])/3]]

middle: -1 - sqrt[7/3] cos[1/3 arctan[(2 sqrt[79/3])/3]] + sqrt[7] sin[1/3 arctan[(2 sqrt[79/3])/3]]

least: -1 - sqrt[7/3] cos[1/3 arctan[(2 sqrt[79/3])/3]] - sqrt[7] sin[1/3 arctan[(2 sqrt[79/3])/3]]

Example

greatest root: 1.7784571182583887319...
middle root: -1.2891685464483099691...
least root: -3.4892885718100787628...

Mathematica

a = 1; b = 1; c = 1; u = 0; v = 2; w = 4; d = 1;
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]]]  (* A316137, least *)
RealDigits[u[[2]]]  (* A316138, middle *)
RealDigits[u[[3]]]  (* A316139, greatest *)

Crossrefs

Cf. A305328, A316137, A316139.

Sequence in context: A152748 A158933 A200502 * A011062 A155922 A201896

Adjacent sequences: A316135 A316136 A316137 * A316139 A316140 A316141

Keyword

nonn,cons

Author

Clark Kimberling, Jul 21 2018

Status

approved