A305327 Decimal expansion of the middle x such that 1/x + 1/(x+1) + 1/(x+2) = 1.

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

Offset

0,1

Comments

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

Greatest root: A305326;

Least root: A305328.

Links

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

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. A305326, A305328.

Sequence in context: A199438 A220129 A192039 * A112812 A241624 A159275

Adjacent sequences: A305324 A305325 A305326 * A305328 A305329 A305330

Keyword

nonn,easy,cons

Author

Clark Kimberling, May 30 2018

Status

approved