A316134 Decimal expansion of the least x such that 1/x + 1/(x+2) + 1/(x+3) = 1 (negated).

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

Offset

1,1

Comments

Equivalently, the least root of x^3 + 2*x^2 - 4*x - 6;

Middle root: A316135;

Greatest root: A316136.

See A305328 for a guide to related sequences.

Links

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

Formula

greatest root: -2/3 + (8/3)*cos((1/3)*arctan((3*sqrt(303))/37))

middle: -2/3 - (4/3)*cos((1/3)*arctan((3*sqrt(303))/37)) + (4*sin((1/3)*arctan((3*sqrt(303))/37)))/sqrt(3)

least: -2/3 - (4/3)*cos((1/3)*arctan((3*sqrt(303))/37)) - (4*sin((1/3)*arctan((3*sqrt(303))/37)))/sqrt(3)

Example

greatest root: 1.8661982625090225055...
middle root: -1.2107558809591917224...
least root: -2.6554423815498307831...

Mathematica

a = 1; b = 1; c = 1; u = 0; v = 2; w = 3; d = 1;
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]]]  (* A316134 *)
RealDigits[u[[2]]]  (* A316135 *)
RealDigits[u[[3]]]  (* A316136 *)

Prog

(PARI) -(-2/3 - (4/3)*cos((1/3)*atan((3*sqrt(303))/37)) - (4*sin((1/3)*atan((3*sqrt(303))/37)))/sqrt(3)) \\ Felix Fröhlich, Jul 07 2018

Crossrefs

Cf. A305328, A316135, A316136.

Sequence in context: A171897 A105029 A309040 * A273621 A190124 A067548

Adjacent sequences: A316131 A316132 A316133 * A316135 A316136 A316137

Keyword

nonn,cons

Author

Clark Kimberling, Jun 28 2018

Status

approved