login
A316256
Decimal expansion of the middle x such that 1/x + 1/(x+2) + 1/(x+4) = 3.
4
1, 6, 8, 3, 7, 6, 1, 8, 3, 6, 6, 7, 8, 0, 3, 4, 3, 1, 2, 9, 0, 6, 6, 5, 2, 5, 9, 4, 2, 5, 1, 7, 0, 2, 6, 1, 6, 4, 7, 6, 3, 3, 7, 0, 8, 9, 7, 9, 4, 2, 7, 6, 1, 5, 3, 6, 1, 4, 9, 2, 7, 3, 0, 0, 2, 9, 0, 8, 2, 1, 5, 3, 6, 3, 3, 6, 9, 2, 6, 8, 6, 1, 0, 2, 3, 9
OFFSET
1,2
COMMENTS
Equivalently, the least root of 3*x^3 + 15*x^2 + 12 x - 8.
Least: A316255;
Greatest: A316257.
See A305328 for a guide to related sequences.
FORMULA
greatest root: -(5/3) + (2/3) sqrt(13) cos((1/3) arctan(6 sqrt(61)))
****
middle: -(5/3) - (1/3) sqrt(13) cos((1/3) arctan(6 sqrt(61))) - sqrt(13/3) sin((1/3) arctan(6 sqrt(61)))
****
least: -(5/3) - (1/3) sqrt(13) cos((1/3) arctan(6 sqrt(61))) + sqrt(13/3) sin((1/3) arctan(6 sqrt(61)))
EXAMPLE
greatest root: 0.4234942709347976489...
middle root: -1.683761836678034312...
least root: -3.739732434256763336...
MATHEMATICA
a = 1; b = 1; c = 1; u = 0; v = 2; w = 4; 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]]] (* A316257, greatest *)
RealDigits[y[[2]]] (* A316255, least *)
RealDigits[y[[3]]] (* A316256, middle *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Sep 14 2018
STATUS
approved