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A316253
Decimal expansion of the middle x such that 1/x + 1/(x+2) + 1/(x+3) = 3.
4
1, 6, 5, 0, 8, 9, 8, 5, 2, 8, 0, 9, 1, 8, 0, 3, 1, 4, 7, 9, 7, 4, 2, 8, 7, 9, 9, 2, 6, 5, 7, 2, 5, 4, 6, 9, 8, 7, 7, 3, 5, 0, 7, 7, 5, 3, 7, 2, 9, 7, 2, 4, 7, 4, 9, 7, 2, 8, 2, 1, 9, 7, 1, 8, 7, 3, 8, 1, 4, 4, 1, 5, 9, 7, 5, 3, 1, 1, 9, 9, 9, 1, 8, 6, 6, 2
OFFSET
1,2
COMMENTS
Equivalently, the least root of 3*x^3 + 12*x^2 + 8 x - 6;
Least: A316252;
Greatest root: A316254.
See A305328 for a guide to related sequences.
FORMULA
greatest root: -(4/3) + (4/3) sqrt(2) cos((1/3) arctan(sqrt(391)/11))
***
middle: -(4/3) - (2/3) sqrt(2) cos((1/3) arctan(sqrt(391)/11)) + 2 sqrt(2/3) sin((1/3) arctan(sqrt(391)/11))
****
least: -(4/3) - (2/3) sqrt(2) cos((1/3) arctan(sqrt(391)/11)) - 2 sqrt(2/3) sin((1/3) arctan(sqrt(391)/11))
EXAMPLE
greatest root: 0.4351172195495135109...
middle root: -1.650898528091803148...
least root: -2.784218691457710362...
MATHEMATICA
a = 1; b = 1; c = 1; u = 0; v = 2; 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]]] (* A316254, greatest *)
RealDigits[y[[2]]] (* A316252, least *)
RealDigits[y[[3]]] (* A316253, middle *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Sep 08 2018
STATUS
approved