A316254 Decimal expansion of the greatest x such that 1/x + 1/(x+2) + 1/(x+3) = 3.

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

Offset

1,1

Comments

Equivalently, the least root of 3*x^3 + 12*x^2 + 8 x - 6;

Least: A316252;

Middle: A316253.

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: -(4/3) + (4/3)*sqrt(2)*cos((1/3)*arctan(sqrt(391)/11))

****

middle: -(4/3) - (2/3)*sqrt(2)*cos((1/3) arctan(sqrt(391)/11)) + 2*sqrt(2/3)*sin((1/3) arctan(sqrt(391)/11))

****

least: -(4/3) - (2/3)*sqrt(2)*cos((1/3) arctan(sqrt(391)/11)) - 2*sqrt(2/3)*sin((1/3) arctan(sqrt(391)/11))

Example

greatest root: 0.4351172195495135109...
middle root: -1.650898528091803148...
least root: -2.784218691457710362...

Mathematica

a = 1; b = 1; c = 1; u = 0; v = 2; w = 3; 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]]] (* A316254, greatest *)
RealDigits[y[[2]]] (* A316252, least *)
RealDigits[y[[3]]] (* A316253, middle *)

Prog

(PARI) polrootsreal(3*x^3 + 12*x^2 + 8*x - 6)[3] \\ Charles R Greathouse IV, May 17 2026

Crossrefs

Cf. A305328, A316252, A316253.

Sequence in context: A343919 A152191 A292612 * A029934 A246665 A274260

Adjacent sequences: A316251 A316252 A316253 * A316255 A316256 A316257

Keyword

nonn,cons

Author

Clark Kimberling, Sep 08 2018

Status

approved