A316257 Decimal expansion of the greatest x such that 1/x + 1/(x+2) + 1/(x+4) = 3.

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

Offset

1,1

Comments

Equivalently, the least root of 3*x^3 + 15*x^2 + 12 x - 8.

Least: A316255

Middle: A316256;

See A305328 for a guide to related sequences.

Links

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

Index entries for algebraic numbers, degree 3.

Formula

greatest root: -(5/3) + (2/3) sqrt(13) cos((1/3) arctan(6 sqrt(61)))

****

middle: -(5/3) - (1/3) sqrt(13) cos((1/3) arctan(6 sqrt(61))) - sqrt(13/3) sin((1/3) arctan(6 sqrt(61)))

****

least: -(5/3) - (1/3) sqrt(13) cos((1/3) arctan(6 sqrt(61))) + sqrt(13/3) sin((1/3) arctan(6 sqrt(61)))

Example

greatest root: 0.4234942709347976489...
middle root: -1.683761836678034312...
least root: -3.739732434256763336...

Mathematica

a = 1; b = 1; c = 1; u = 0; v = 2; w = 4; d = 3;
r[x_] := a/(x + u) + b/(x + v) + c/(x + w);
t = x /. ComplexExpand[Solve[r[x] == d, x]]
N[t, 20]
y = Re[N[t, 200]];
RealDigits[y[[1]]] (* A316257, greatest *)
RealDigits[y[[2]]] (* A316255, least *)
RealDigits[y[[3]]] (* A316256, middle *)

Prog

(PARI) polrootsreal(3*x^3 + 15*x^2 + 12*x - 8)[3] \\ Charles R Greathouse IV, May 17 2026

Crossrefs

Cf. A305328, A316255, A316256.

Sequence in context: A225001 A128011 A034927 * A274791 A200024 A247206

Adjacent sequences: A316254 A316255 A316256 * A316258 A316259 A316260

Keyword

nonn,cons

Author

Clark Kimberling, Sep 14 2018

Status

approved