A316259 - OEIS

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A316259

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

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

Least:: A316258

Greatest: A316260;

See A305328 for a guide to related sequences.

LINKS

Muniru A Asiru, Table of n, a(n) for n = 1..1000

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, A316258, A316260.

Sequence in context: A330541 A320575 A110218 * A057892 A334188 A265993

Adjacent sequences: A316256 A316257 A316258 * A316260 A316261 A316262

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Sep 14 2018

EXTENSIONS

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

STATUS

approved