A305328 Decimal expansion of the least x such that 1/x + 1/(x+1) + 1/(x+2) = 1 (negated).

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

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-8 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.

Index entries for algebraic numbers, degree 3

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 A355336

Adjacent sequences: A305325 A305326 A305327 * A305329 A305330 A305331

Keyword

nonn,easy,cons

Author

Clark Kimberling, May 30 2018

Status

approved