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A305328 Decimal expansion of the least x such that 1/x + 1/(x+1) + 1/(x+2) = 1 (negated). 36
1, 6, 7, 5, 1, 3, 0, 8, 7, 0, 5, 6, 6, 6, 4, 6, 0, 7, 0, 8, 8, 9, 6, 2, 1, 7, 9, 8, 1, 5, 0, 0, 6, 0, 4, 8, 0, 8, 0, 8, 0, 3, 2, 5, 2, 7, 6, 7, 7, 3, 7, 2, 7, 3, 2, 6, 1, 2, 1, 5, 3, 8, 6, 9, 8, 4, 1, 4, 4, 2, 0, 4, 2, 9, 9, 0, 4, 9, 9, 3, 1, 9, 7, 4, 2, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Equivalently, the least root of x^3 - 4*x - 2;

Middle root:  A305327;

Greatest root:  A305326.

From Clark Kimberling, Sep 03 2018: (Start)

The following guide applies to zeros of rational functions of the form 1/x + 1/(x+v) + 1/(x+w) = d, for selected values of v,w, and d. The three zeros are distinct real numbers, denoted as least, middle, and greatest.  These zeros are also the roots of the following cubic polynomial: p(u,v,w,d) = d x^3 + (d v + d w - 3) x^2 + (d v w - 2 v - 2 w) x - v w.

  v  w  d  p(u,v,w,d)        least    middle   greatest

  1  2  1  x^3-4x-2          A305328  A305327  A305326

  1  3  1  x^3+x^2-5x-3      A316131  A316132  A316133

  2  3  1  x^3+2x^2-4x-6     A316134  A316135  A316136

  2  4  1  x^3+3x^2-4x-8     A316137  A316138  A316139

  1  2  2  2x^3+3x^2-2x-2    A316161  A316162  A316163

  1  3  2  2x^3+5x^2-2x-3    A316164  A316165  A316166

  2  4  2  2x^3+9x^2-4x-3    A316167  A316168  A316169

  1  2  3  3x^3+6x^2-2       A316246  A316247  A316248

  1  3  3  3x^3+9x^2+x-3     A316249  A316250  A316251

  2  3  3  3x^3+12x^2+8x-6   A316252  A316253  A316254

  2  4  3  3x^3+15x^2+12x-8  A316255  A316256  A316257

  3  4  3  3x^3+18x^2+22x-12 A316258  A316259  A316260

(End)

LINKS

Table of n, a(n) for n=1..86.

FORMULA

greatest:  (4*cos((1/3)*arctan(sqrt(37/3)/3)))/sqrt(3);

middle: -((2*cos((1/3)*arctan(sqrt(37/3)/3)))/sqrt(3)) + 2*sin((1/3)*arctan(sqrt(37/3)/3));

least: -((2*cos((1/3)*arctan(sqrt(37/3)/3)))/sqrt(3)) - 2*sin((1/3)*arctan(sqrt(37/3)/3)).

EXAMPLE

greatest root: 2.214319743377535187...

middle root: -0.539188872810889116...

least root: -1.67513087056664607088...

MATHEMATICA

r[x_] := 1/x + 1/(x + 1) + 1/(x + 2);

-Numerator[Factor[r[x] - 1]]

t = x /. ComplexExpand[Solve[r[x] == 1, x]]

u = N[t, 120]

RealDigits[u[[1]]]  (* A305326, greatest root *)

RealDigits[u[[2]]]  (* A305327, middle root *)

RealDigits[u[[3]]]  (* A305328, least root *)

PROG

(PARI) solve(x=-2, -1, x^3 - 4*x - 2) \\ Michel Marcus, Jul 16 2018

CROSSREFS

Cf. A305326, A305327.

Sequence in context: A111969 A296480 A021601 * A332396 A100124 A190648

Adjacent sequences:  A305325 A305326 A305327 * A305329 A305330 A305331

KEYWORD

nonn,easy,cons

AUTHOR

Clark Kimberling, May 30 2018

STATUS

approved

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Last modified January 24 19:12 EST 2021. Contains 340411 sequences. (Running on oeis4.)