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A316165
Decimal expansion of the middle x such that 1/x + 1/(x+1) + 1/(x+3) = 2.
4
6, 7, 2, 8, 3, 3, 2, 4, 6, 5, 5, 3, 1, 6, 6, 6, 0, 7, 9, 8, 8, 4, 6, 6, 0, 9, 4, 2, 3, 6, 6, 1, 6, 0, 6, 3, 8, 2, 1, 9, 8, 0, 1, 8, 5, 6, 5, 6, 7, 6, 2, 1, 1, 6, 0, 1, 8, 2, 6, 7, 1, 5, 5, 8, 1, 7, 4, 9, 1, 6, 1, 9, 7, 0, 7, 0, 9, 6, 1, 8, 3, 7, 1, 5, 4, 4
OFFSET
0,1
COMMENTS
Equivalently, the least root of 2*x^3 + 5*x^2 - 2*x - 3;
Middle root: A316165;
Greatest root: A316166.
See A305328 for a guide to related sequences.
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 *)
PROG
(PARI) solve(x=-1, 0, 2*x^3+5*x^2-2*x-3) \\ Altug Alkan, Aug 27 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Aug 08 2018
STATUS
approved