A316163 Decimal expansion of the greatest x such that 1/x + 1/(x+1) + 1/(x+2) = 2.

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

Offset

1,1

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=1..86.

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, A316162.

Sequence in context: A255287 A257342 A252850 * A346908 A195717 A309930

Adjacent sequences: A316160 A316161 A316162 * A316164 A316165 A316166

Keyword

nonn,cons

Author

Clark Kimberling, Aug 08 2018

Status

approved