A316139 - OEIS

A316139

Decimal expansion of the greatest x such that 1/x + 1/(x+2) + 1/(x+4) = 1.

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

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OFFSET

1,2

COMMENTS

Equivalently, the greatest root of x^3 + 3*x^2 - 4*x - 8;

Least root: A316137

Middle root: A316138;

Greatest root: A316139.

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 - sqrt[7/3] cos[1/3 arctan[(2 sqrt[79/3])/3]] + sqrt[7] sin[1/3 arctan[(2 sqrt[79/3])/3]]

least: -1 - sqrt[7/3] cos[1/3 arctan[(2 sqrt[79/3])/3]] - sqrt[7] sin[1/3 arctan[(2 sqrt[79/3])/3]]

EXAMPLE

greatest root: 1.7784571182583887319...

middle root: -1.2891685464483099691...

least root: -3.4892885718100787628...

MATHEMATICA

a = 1; b = 1; c = 1; u = 0; v = 2; w = 4; 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]]] (* A316137, least *)

RealDigits[u[[2]]] (* A316138, middle *)

RealDigits[u[[3]]] (* A316139, greatest *)

CROSSREFS

Cf. A305328, A316137, A316138.

Sequence in context: A019718 A374837 A353973 * A198992 A201668 A021853

Adjacent sequences: A316136 A316137 A316138 * A316140 A316141 A316142

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Jul 21 2018

STATUS

approved