A316162 Decimal expansion of the middle x such that 1/x + 1/(x+1) + 1/(x+2) = 2.

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

Offset

0,1

Comments

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

Middle root: A316162;

Greatest root: A316163.

See A305328 for a guide to related sequences.

Links

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

Formula

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

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

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

Example

greatest root: 0.88922855912919436594...
middle root: -0.64458427322415498454...
least root: -1.7446442859050393814...

Mathematica

a = 1; b = 1; c = 1; u = 0; v = 1; w = 2; 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]]]  (* A316161, least *)
RealDigits[u[[2]]]  (* A316162, middle *)
RealDigits[u[[3]]]  (* A316163, greatest *)

Crossrefs

Cf. A305328, A316161, A316163.

Sequence in context: A317127 A195359 A338303 * A332431 A198840 A211268

Adjacent sequences: A316159 A316160 A316161 * A316163 A316164 A316165

Keyword

nonn,cons

Author

Clark Kimberling, Jul 21 2018

Status

approved