A316246 Decimal expansion of the least x such that 1/x + 1/(x+1) + 1/(x+2) = 3.

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

Offset

1,2

Comments

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

Middle root: A316247;

Greatest root: A316248.

See A305328 for a guide to related sequences.

Links

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

Formula

greatest root: -2/3 + (4/3)*cos((1/3)*arctan(3*sqrt(7)))

****

middle: -2/3 - (2/3)*cos((1/3)*arctan(3*sqrt(7))) + 2*sin((1/3)*arctan(3*sqrt(7)))/sqrt(3)

****

least: -2/3 - (2/3)*cos((1/3)*arctan(3*sqrt(7))) - 2*sin((1/3)*arctan(3*sqrt(7)))/sqrt(3)

Example

greatest root: 0.5148689384387165869...
middle root: -0.7223517244643762951...
least root: -1.792517213974340291...

Mathematica

a = 1; b = 1; c = 1; u = 0; v = 1; w = 2; d = 3;
r[x_] := a/(x + u) + b/(x + v) + c/(x + w);
t = x /. ComplexExpand[Solve[r[x] == d, x]]
N[t, 20]
u = N[t, 200];
RealDigits[u[[1]]]  (* A316246, greatest *)
RealDigits[u[[2]]]  (* A316247, least *)
RealDigits[u[[3]]]  (* A316248, middle *)
RealDigits[Root[1/x+1/(x+1)+1/(x+2)-3, 1], 10, 120][[1]] (* Harvey P. Dale, May 28 2023 *)

Crossrefs

Cf. A305328, A316247, A316248.

Sequence in context: A154197 A094148 A154215 * A249546 A336076 A369235

Adjacent sequences: A316243 A316244 A316245 * A316247 A316248 A316249

Keyword

nonn,cons

Author

Clark Kimberling, Aug 19 2018

Extensions

a(86) corrected by Andrew Howroyd, Nov 04 2018

Status

approved