A316247 Decimal expansion of the middle x such that 1/x + 1/(x+1) + 1/(x+2) = 3.

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

Offset

0,1

Comments

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

Least root: A316246;

Greatest root: A316248.

See A305328 for a guide to related sequences.

Links

Table of n, a(n) for n=0..85.

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 *)

Crossrefs

Cf. A305328, A316246, A316248.

Sequence in context: A195723 A138996 A010141 * A299922 A199370 A334961

Adjacent sequences: A316244 A316245 A316246 * A316248 A316249 A316250

Keyword

nonn,cons

Author

Clark Kimberling, Aug 19 2018

Status

approved