A316260 Decimal expansion of the greatest x such that 1/x + 1/(x+3) + 1/(x+4) = 3.

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

Offset

0,1

Comments

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

Least:: A316258

Middle: A316259;

See A305328 for a guide to related sequences.

Links

Muniru A Asiru, Table of n, a(n) for n = 0..2000

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...

Maple

evalf(solve(3*x^3+18*x^2+22*x-12=0, x)[1], 90); # Muniru A Asiru, Oct 07 2018

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, A316259.

Sequence in context: A195775 A363728 A362562 * A119708 A106142 A092512

Adjacent sequences: A316257 A316258 A316259 * A316261 A316262 A316263

Keyword

nonn,cons

Author

Clark Kimberling, Sep 14 2018

Status

approved