A305326 Decimal expansion of the greatest x such that 1/x + 1/(x+1) + 1/(x+2) = 1.

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

Offset

1,1

Comments

Equivalently, the greatest root of x^3 - 4*x - 2;

Middle root: A305327;

Least root: A305328.

Links

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

Formula

greatest: (4 cos[1/3 arctan[sqrt[37/3]/3]])/sqrt[3]

middle:

-((2 cos[1/3 arctan[sqrt[37/3]/3]])/sqrt[3]) + 2 sin[1/3 arctan[sqrt[37/3]/3]]

least:

-((2 cos[1/3 arctan[sqrt[37/3]/3]])/sqrt[3]) - 2 sin[1/3 arctan[sqrt[37/3]/3]]

Example

greatest root: 2.214319743377535187...
middle root: -0.539188872810889116...
least root: -1.67513087056664607088...

Mathematica

r[x_] := 1/x + 1/(x + 1) + 1/(x + 2);
-Numerator[Factor[r[x] - 1]]
t = x /. ComplexExpand[Solve[r[x] == 1, x]]
u = N[t, 120]
RealDigits[u[[1]]]  (* A305326 *)
RealDigits[u[[2]]]  (* A305327 *)
RealDigits[u[[3]]]  (* A305328 *)

Crossrefs

Cf. A305327, A305328.

Sequence in context: A197376 A113072 A328025 * A122918 A177424 A286332

Adjacent sequences: A305323 A305324 A305325 * A305327 A305328 A305329

Keyword

nonn,easy,cons

Author

Clark Kimberling, May 30 2018

Status

approved