A316166 Decimal expansion of the greatest x such that 1/x + 1/(x+1) + 1/(x+3) = 2.

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

Offset

0,1

Comments

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

Middle root: A316165;

Greatest root: A316166.

See A305328 for a guide to related sequences.

Links

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

Formula

greatest root: -(5/6) + (1/6) sqrt(37) cos((1/3)(Pi - arctan((6 sqrt(1329))/53))) + (1/6) sqrt(37) cos((1/3)(-Pi + arctan((6 sqrt(1329))/53)))

****

middle: -(5/6) - (1/12) sqrt(37) cos((1/3)(Pi - arctan((6 sqrt(1329))/53))) -

(1/12) sqrt(37) cos((1/3)(-Pi + arctan((6 sqrt(1329))/53))) +

(1/4) sqrt(37/3) sin((1/3)(Pi - arctan((6 sqrt(1329))/53))) -

(1/4) sqrt(37/3) sin((1/3)(-Pi + arctan((6 sqrt(1329))/53)))

****

least: -(5/6) - (1/12) sqrt(37) cos((1/3)(Pi - arctan((6 sqrt(1329))/53))) -

(1/12) sqrt(37) cos(1/3(-Pi + arctan((6 sqrt(1329))/53))) -

(1/4) sqrt(37/3) sin((1/3)(Pi - arctan((6 sqrt(1329))/53))) +

(1/4) sqrt(37/3) sin(1/3)(-Pi + arctan((6 sqrt(1329))/53)))

Example

greatest root: 0.83684889130097120054...
middle root: -0.67283324655316660799...
least root: -2.6640156447478045925...

Mathematica

a = 1; b = 1; c = 1; u = 0; v = 1; w = 3; d = 2;
r[x_] := a/(x + u) + b/(x + v) + c/(x + w);
t = Re[x /. ComplexExpand[Solve[r[x] == d, x]]]
N[t, 20]
u = N[t, 200];
u1 = RealDigits[u[[1]]]  (* A316166, greatest *)
u2 = RealDigits[u[[2]]]  (* A316164, least *)
u3 = RealDigits[u[[3]]]  (* A316165, middle *)
RealDigits[Root[1/x+1/(x+1)+1/(x+3)-2, 3], 10, 120][[1]] (* Harvey P. Dale, Sep 06 2022 *)

Crossrefs

Cf. A305328, A316164, A316165.

Sequence in context: A258991 A132035 A153813 * A307106 A372857 A277775

Adjacent sequences: A316163 A316164 A316165 * A316167 A316168 A316169

Keyword

nonn,cons

Author

Clark Kimberling, Aug 08 2018

Status

approved