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A316132 Decimal expansion of the middle x such that 1/x + 1/(x+1) + 1/(x+3) = 1, negated. 4
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 (list; constant; graph; refs; listen; history; text; internal format)
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.

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

CROSSREFS

Cf. A305328, A316131, A316133.

Sequence in context: A343480 A251735 A232811 * A261159 A145737 A108763

Adjacent sequences:  A316129 A316130 A316131 * A316133 A316134 A316135

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Jun 27 2018

STATUS

approved

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Last modified December 8 01:25 EST 2021. Contains 349590 sequences. (Running on oeis4.)