A316258 Decimal expansion of the least x such that 1/x + 1/(x+3) + 1/(x+4) = 3 (negated).

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

Offset

1,1

Comments

Equivalently, the least root of 3*x^3 + 18*x^2 + 22 x - 12.

Middle:: A316259

Greatest: A316260;

See A305328 for a guide to related sequences.

Links

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

Formula

greatest root: -2 + (2/3) sqrt(14) cos((1/3) arctan(sqrt(181/2)/9))

****

middle: -2 - (1/3)sqrt(14) cos((1/3) arctan(sqrt(181/2)/9)) + sqrt(14/3) sin((1/3) arctan(sqrt(181/2)/9))

****

least: -2 - 1/3 sqrt(14) cos((1/3) arctan(sqrt(181/2)/9)) - sqrt(14/3) sin((1/3) arctan(sqrt(181/2)/9))

Example

greatest root: 0.4033761543003640184...
middle root: -2.623324901998131634... [Corrected by A.H.M. Smeets, Sep 17 2018]
least root: -3.780051252302232384...

Mathematica

a = 1; b = 1; c = 1; u = 0; v = 3; 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]]]  (* A316259, middle *)
RealDigits[y[[2]]]  (* A316258, least *)
RealDigits[y[[3]]]  (* A316260, greatest *)

Crossrefs

Cf. A305328, A316259, A316260.

Sequence in context: A272981 A086839 A359322 * A258405 A064208 A078004

Adjacent sequences: A316255 A316256 A316257 * A316259 A316260 A316261

Keyword

nonn,cons

Author

Clark Kimberling, Sep 14 2018

Extensions

Name corrected by A.H.M. Smeets, Sep 17 2018

Status

approved