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A316161
Decimal expansion of the least x such that 1/x + 1/(x+1) + 1/(x+2) = 2.
4
1, 7, 4, 4, 6, 4, 4, 2, 8, 5, 9, 0, 5, 0, 3, 9, 3, 8, 1, 3, 9, 6, 4, 6, 8, 2, 6, 5, 2, 2, 7, 4, 2, 4, 6, 2, 0, 5, 8, 4, 0, 3, 2, 9, 1, 9, 7, 4, 1, 4, 9, 6, 5, 5, 7, 7, 6, 8, 2, 8, 3, 2, 2, 7, 5, 8, 5, 3, 3, 7, 4, 6, 7, 0, 7, 1, 3, 0, 8, 2, 0, 9, 6, 7, 1, 7
OFFSET
0,2
COMMENTS
Equivalently, the least root of 2*x^3 + 3*x^2 - 2*x - 2;
Middle root: A316162;
Greatest root: A316163.
See A305328 for a guide to related sequences.
FORMULA
greatest root: -(1/2) + sqrt[7/3] cos[1/3 arctan[(2 sqrt[79/3])/3]]
middle: -(1/2) - 1/2 sqrt[7/3] cos[1/3 arctan[(2 sqrt[79/3])/3]] + 1/2 sqrt[7] sin[1/3 arctan[(2 sqrt[79/3])/3]]
least: -(1/2) - 1/2 sqrt[7/3] cos[1/3 arctan[(2 sqrt[79/3])/3]] - 1/2 sqrt[7] sin[1/3 arctan[(2 sqrt[79/3])/3]]
EXAMPLE
greatest root: 0.88922855912919436594...
middle root: -0.64458427322415498454...
least root: -1.7446442859050393814...
MATHEMATICA
a = 1; b = 1; c = 1; u = 0; v = 1; w = 2; 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]]] (* A316161, least *)
RealDigits[u[[2]]] (* A316162, middle *)
RealDigits[u[[3]]] (* A316163, greatest *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Jul 21 2018
STATUS
approved