A316161 Decimal expansion of the least x such that 1/x + 1/(x+1) + 1/(x+2) = 2.

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

Offset

0,2

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.

Index entries for algebraic numbers, degree 3.

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 *)

Prog

(PARI) polrootsreal(2*x^3 - 3*x^2 - 2*x + 2)[1] \\ Charles R Greathouse IV, Apr 19 2026

Crossrefs

Cf. A305328, A316162, A316163.

Sequence in context: A245074 A194474 A348736 * A377010 A395847 A153349

Adjacent sequences: A316158 A316159 A316160 * A316162 A316163 A316164

Keyword

nonn,cons

Author

Clark Kimberling, Jul 21 2018

Status

approved