A316134 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

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

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

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