A316135 Decimal expansion of the middle x such that 1/x + 1/(x+2) + 1/(x+3) = 1 (negated).

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

Offset

1,2

Comments

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

Least root: A316134

Middle root: A316135;

Greatest root: A316136.

See A305328 for a guide to related sequences.

Links

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

Index entries for algebraic numbers, degree 3.

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, least *)
RealDigits[u[[2]]]  (* A316135, middle *)
RealDigits[u[[3]]]  (* A316136, greatest *)

Prog

(PARI) polrootsreal(x^3 + 2*x^2 - 4*x - 6)[2] \\ Charles R Greathouse IV, May 17 2026

Crossrefs

Cf. A305328, A316134, A316136.

Sequence in context: A109971 A357585 A284797 * A327620 A325872 A380369

Adjacent sequences: A316132 A316133 A316134 * A316136 A316137 A316138

Keyword

nonn,cons

Author

Clark Kimberling, Jul 21 2018

Status

approved