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A316168
Decimal expansion of the middle x such that 1/x + 1/(x+2) + 1/(x+4) = 2, negated.
4
1, 5, 5, 2, 6, 6, 2, 3, 2, 6, 2, 1, 3, 5, 2, 6, 0, 6, 1, 7, 8, 0, 6, 9, 9, 0, 8, 4, 1, 1, 0, 3, 0, 6, 0, 3, 6, 8, 2, 3, 4, 3, 5, 8, 2, 1, 7, 8, 7, 4, 8, 1, 1, 9, 0, 2, 5, 4, 3, 1, 8, 2, 8, 1, 8, 8, 1, 1, 7, 6, 0, 9, 9, 7, 5, 4, 3, 5, 6, 4, 4, 7, 6, 2, 2, 8
OFFSET
1,2
COMMENTS
Equivalently, the least root of 2*x^3 + 9*x^2 + 4*x - 8;
Middle root: A316168;
Greatest root: A316169.
See A305328 for a guide to related sequences.
FORMULA
greatest root: -(3/2) + sqrt(19/3) cos(1/3 arctan((4 sqrt(427/3))/3))
middle root: -(3/2) - 1/2 sqrt(19/3) cos(1/3 arctan((4 sqrt(427/3))/3)) + 1/2 sqrt(19) sin(1/3 arctan((4 sqrt(427/3))/3))
least root: -(3/2) - 1/2 sqrt(19/3) cos(1/3 arctan((4 sqrt(427/3))/3)) - 1/2 sqrt(19) sin(1/3 arctan((4 sqrt(427/3))/3))
EXAMPLE
greatest root: 0.70530340009105630377...
middle root: -1.5526623262135260618...
least root: -3.6526410738775302420...
MATHEMATICA
a = 1; b = 1; c = 1; u = 0; v = 2; w = 4; d = 2;
r[x_] := a/(x + u) + b/(x + v) + c/(x + w);
t = x /. ComplexExpand[Solve[r[x] == d, x]]
N[t, 20]
u = N[t, 200];
RealDigits[u[[1]]] (* A316167, greatest *)
RealDigits[u[[2]]] (* A316168, middle *)
RealDigits[u[[3]]] (* A316169, least *)
PROG
(PARI) solve(x=-2, -1, 2*x^3 + 9*x^2 + 4*x - 8) \\ Michel Marcus, Aug 11 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Aug 09 2018
STATUS
approved