A316250 - OEIS

%I #7 Sep 13 2018 08:05:10

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

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

%U 8,9,2,8,2,4,7,2,3,5,7,6,1,2,9,1,5,8

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

%C Equivalently, the middle root of 3*x^3 + 9*x^2 + x - 3;

%C Least root: A316249;

%C Greatest root: A316251.

%C See A305328 for a guide to related sequences.

%F greatest root: -1 + (2/3)*sqrt(2)*cos(Pi/3 - (1/3)*arctan(sqrt(431)/9))) + (2/3)*sqrt(2)*cos(-Pi/3 + (1/3)*arctan(sqrt(431)/9)))

%F ****

%F middle: -1 - (1/3)*sqrt(2)*cos(Pi/3 - (1/3)*arctan(sqrt(431)/9))) - (1/3)*sqrt(2)*cos(-Pi/3 + (1/3)*arctan(sqrt(431)/9))) + sqrt(2/3)*sin(Pi/3 - (1/3)*arctan(sqrt(431)/9))) - sqrt(2/3)*sin(-Pi/3 + (1/3)*arctan(sqrt(431)/9)))

%F ****

%F least: -1 - (1/3)*sqrt(2)*cos(Pi/3 - (1/3)*arctan(sqrt(431)/9))) - (1/3)*sqrt(2)*cos(-Pi/3 + (1/3)*arctan(sqrt(431)/9))) - sqrt(2/3)*sin(Pi/3 - (1/3)*arctan(sqrt(431)/9))) + sqrt(2/3)*sin(-Pi/3 + (1/3)*arctan(sqrt(431)/9)))

%e greatest root: 0.4896787901169438959...

%e middle root: -0.7436853143229163734...

%e least root: -2.745993475794027522...

%t a = 1; b = 1; c = 1; u = 0; v = 1; w = 3; 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]]] (* A316251, greatest *)

%t RealDigits[y[[2]]] (* A316249, least *)

%t RealDigits[y[[3]]] (* A316250, middle *)

%Y Cf. A305328, A316249, A316251.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Aug 21 2018