

A316255


Decimal expansion of the least x such that 1/x + 1/(x+2) + 1/(x+4) = 3.


4



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

1,1


COMMENTS

Equivalently, the least root of 3*x^3 + 15*x^2 + 12 x  8.
Middle: A316256;
Greatest: A316257.
See A305328 for a guide to related sequences.


LINKS

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


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

Cf. A305328, A316256, A316257.
Sequence in context: A074176 A005596 A159566 * A096385 A205723 A088837
Adjacent sequences: A316252 A316253 A316254 * A316256 A316257 A316258


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Sep 14 2018


STATUS

approved



