|
%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
|
