A316255 - OEIS

%I #5 Sep 16 2018 21:44:19

%S 3,7,3,9,7,3,2,4,3,4,2,5,6,7,6,3,3,3,6,0,7,3,7,0,8,4,2,3,3,3,9,6,8,3,

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

%U 4,5,2,2,8,2,5,6,1,9,0,0,8,3,2,8,5,9

%N Decimal expansion of the least x such that 1/x + 1/(x+2) + 1/(x+4) = 3.

%C Equivalently, the least root of 3*x^3 + 15*x^2 + 12 x - 8.

%C Middle: A316256;

%C Greatest: A316257.

%C See A305328 for a guide to related sequences.

%F greatest root: -(5/3) + (2/3) sqrt(13) cos((1/3) arctan(6 sqrt(61)))

%F ****

%F middle: -(5/3) - (1/3) sqrt(13) cos((1/3) arctan(6 sqrt(61))) - sqrt(13/3) sin((1/3) arctan(6 sqrt(61)))

%F ****

%F least: -(5/3) - (1/3) sqrt(13) cos((1/3) arctan(6 sqrt(61))) + sqrt(13/3) sin((1/3) arctan(6 sqrt(61)))

%e greatest root: 0.4234942709347976489...

%e middle root: -1.683761836678034312...

%e least root: -3.739732434256763336...

%t a = 1; b = 1; c = 1; u = 0; v = 2; w = 4; d = 3;

%t r[x_] := a/(x + u) + b/(x + v) + c/(x + w);

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

%t N[t, 20]

%t y = Re[N[t, 200]];

%t RealDigits[y[[1]]] (* A316257, greatest *)

%t RealDigits[y[[2]]] (* A316255, least *)

%t RealDigits[y[[3]]] (* A316256, middle *)

%Y Cf. A305328, A316256, A316257.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Sep 14 2018