A316249 Decimal expansion of the least x such that 1/x + 1/(x+1) + 1/(x+3) = 3.

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

Offset

1,1

Comments

Equivalently, the least root of 3*x^3 + 9*x^2 + x - 3;

Middle root: A316250;

Greatest root: A316251.

See A305328 for a guide to related sequences.

Links

Table of n, a(n) for n=1..86.

Formula

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)))

****

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)))

****

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)))

Example

greatest root: 0.4896787901169438959...
middle root: -0.7436853143229163734...
least root: -2.745993475794027522...

Mathematica

a = 1; b = 1; c = 1; u = 0; v = 1; w = 3; d = 3;
r[x_] := a/(x + u) + b/(x + v) + c/(x + w);
t = x /. ComplexExpand[Solve[r[x] == d, x]]
N[t, 20]
y = Re[N[t, 200]];
RealDigits[y[[1]]] (* A316251, greatest *)
RealDigits[y[[2]]] (* A316249, least *)
RealDigits[y[[3]]] (* A316250, middle *)

Crossrefs

Cf. A305328, A316250, A316251.

Sequence in context: A347767 A325644 A198036 * A197143 A132724 A306579

Adjacent sequences: A316246 A316247 A316248 * A316250 A316251 A316252

Keyword

nonn,cons

Author

Clark Kimberling, Aug 21 2018

Status

approved