A316132 Decimal expansion of the middle x such that 1/x + 1/(x+1) + 1/(x+3) = 1, negated.

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

Offset

0,1

Comments

Equivalently, the middle root of x^3 + x^2 - 5*x - 3;

Least root: A316131;

Greatest root: A316133.

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: -1/3 + (8/3)*cos((1/3)*arctan((9*sqrt(47))/17))

middle: -1/3 - (4/3)*cos((1/3)*arctan((9*sqrt(47))/17)) + (4*sin((1/3)*arctan((9*sqrt(47))/17)))/sqrt(3)

least: -1/3 - (4/3)*cos((1/3)*arctan((9*sqrt(47))/17)) - (4*sin((1/3)*arctan((9*sqrt(47))/17)))/sqrt(3)

Example

greatest root: 2.0861301976514940912...
middle root: -0.57199326831620301856...
least root: -2.5141369293352910727...

Mathematica

a = 1; b = 1; c = 1; u = 0; v = 1; w = 3; 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]]]  (* A316131 *)
RealDigits[u[[2]]]  (* A316132 *)
RealDigits[u[[3]]]  (* A316133 *)

Prog

(PARI) solve(x=-1, 0, x^3+x^2-5*x-3) \\ Jianing Song, Aug 01 2018
(PARI) polrootsreal(x^3 - x^2 - 5*x + 3)[2] \\ Charles R Greathouse IV, Apr 10 2026

Crossrefs

Cf. A305328, A316131, A316133.

Sequence in context: A251735 A232811 A380940 * A261159 A145737 A108763

Adjacent sequences: A316129 A316130 A316131 * A316133 A316134 A316135

Keyword

nonn,cons

Author

Clark Kimberling, Jun 27 2018

Status

approved