A316132 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

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

(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