A305327 - OEIS

%I #11 Jul 30 2018 13:15:13

%S 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,

%T 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,

%U 6,0,0,3,4,4,2,7,6,6,9,5,5,0,5,3,7,6

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

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

%C Greatest root:  A305326;

%C Least root:  A305328.

%F greatest:  (4*cos((1/3)*arctan(sqrt(37/3)/3)))/sqrt(3);

%F middle: -((2*cos((1/3)*arctan(sqrt(37/3)/3)))/sqrt(3)) + 2*sin((1/3)*arctan(sqrt(37/3)/3));

%F least: -((2*cos((1/3)*arctan(sqrt(37/3)/3)))/sqrt(3)) - 2*sin((1/3)*arctan(sqrt(37/3)/3)).

%e greatest root: 2.214319743377535187...

%e middle root: -0.539188872810889116...

%e least root: -1.67513087056664607088...

%t r[x_] := 1/x + 1/(x + 1) + 1/(x + 2);

%t -Numerator[Factor[r[x] - 1]]

%t t = x /. ComplexExpand[Solve[r[x] == 1, x]]

%t u = N[t, 120]

%t RealDigits[u[[1]]]  (* A305326 *)

%t RealDigits[u[[2]]]  (* A305327 *)

%t RealDigits[u[[3]]]  (* A305328 *)

%Y Cf. A305326, A305328.

%K nonn,easy,cons

%O 0,1

%A _Clark Kimberling_, May 30 2018