A316131 Decimal expansion of the least x such that 1/x + 1/(x+1) + 1/(x+3) = 1, negated.

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

Offset

1,1

Comments

Equivalently, the least root of x^3 + x^2 - 5*x - 3;

Middle root: A316132;

Greatest root: A316133.

See A305328 for a guide to related sequences.

Links

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

Formula

greatest root: -(1/3) + 8/3 Cos[1/3 ArcTan[(9 Sqrt[47])/17]]

middle: -(1/3) - 4/3 Cos[1/3 ArcTan[(9 Sqrt[47])/17]] + (4 Sin[1/3 ArcTan[(9 Sqrt[47])/17]])/Sqrt[3]

least: -(1/3) - 4/3 Cos[1/3 ArcTan[(9 Sqrt[47])/17]] - (4 Sin[1/3 ArcTan[(9 Sqrt[47])/17]])/Sqrt[3]

Example

greatest root: 2.0861301976514940912...
middle root: -0.57199326831620301856...
least root: -2.5141369293352910727...

Mathematica

a = 1; b = 1; c = 1; u = 0; v = 1; 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]]]  (* A316131 *)
RealDigits[u[[2]]]  (* A316132 *)
RealDigits[u[[3]]]  (* A316133 *)
RealDigits[-x/.FindRoot[1/x+1/(x+1)+1/(x+3)==1, {x, -2.5}, WorkingPrecision -> 120]][[1]] (* Harvey P. Dale, Jun 27 2021 *)

Prog

(PARI) solve(x=-3, -2, x^3+x^2-5*x-3) \\ Jianing Song, Aug 01 2018

Crossrefs

Cf. A305328, A316131, A316132.

Sequence in context: A121428 A239969 A105686 * A153726 A229339 A274880

Adjacent sequences: A316128 A316129 A316130 * A316132 A316133 A316134

Keyword

nonn,cons

Author

Clark Kimberling, Jun 26 2018

Status

approved