OFFSET
0,1
COMMENTS
FORMULA
greatest root: -(5/6) + (1/6) sqrt(37) cos((1/3)(Pi - arctan((6 sqrt(1329))/53))) + (1/6) sqrt(37) cos((1/3)(-Pi + arctan((6 sqrt(1329))/53)))
****
middle: -(5/6) - (1/12) sqrt(37) cos((1/3)(Pi - arctan((6 sqrt(1329))/53))) -
(1/12) sqrt(37) cos((1/3)(-Pi + arctan((6 sqrt(1329))/53))) +
(1/4) sqrt(37/3) sin((1/3)(Pi - arctan((6 sqrt(1329))/53))) -
(1/4) sqrt(37/3) sin((1/3)(-Pi + arctan((6 sqrt(1329))/53)))
****
least: -(5/6) - (1/12) sqrt(37) cos((1/3)(Pi - arctan((6 sqrt(1329))/53))) -
(1/12) sqrt(37) cos(1/3(-Pi + arctan((6 sqrt(1329))/53))) -
(1/4) sqrt(37/3) sin((1/3)(Pi - arctan((6 sqrt(1329))/53))) +
(1/4) sqrt(37/3) sin(1/3)(-Pi + arctan((6 sqrt(1329))/53)))
EXAMPLE
greatest root: 0.83684889130097120054...
middle root: -0.67283324655316660799...
least root: -2.6640156447478045925...
MATHEMATICA
a = 1; b = 1; c = 1; u = 0; v = 1; w = 3; d = 2;
r[x_] := a/(x + u) + b/(x + v) + c/(x + w);
t = Re[x /. ComplexExpand[Solve[r[x] == d, x]]]
N[t, 20]
u = N[t, 200];
u1 = RealDigits[u[[1]]] (* A316166, greatest *)
u2 = RealDigits[u[[2]]] (* A316164, least *)
u3 = RealDigits[u[[3]]] (* A316165, middle *)
RealDigits[Root[1/x+1/(x+1)+1/(x+3)-2, 3], 10, 120][[1]] (* Harvey P. Dale, Sep 06 2022 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Aug 08 2018
STATUS
approved