A316166 - OEIS

%I #12 Sep 06 2022 12:01:40

%S 8,3,6,8,4,8,8,9,1,3,0,0,9,7,1,2,0,0,5,3,5,2,1,4,5,2,2,2,8,2,0,5,6,2,

%T 8,5,9,6,5,9,2,2,0,0,4,0,1,1,3,2,9,1,2,8,3,4,0,4,8,5,2,0,0,6,0,1,1,5,

%U 0,1,1,9,2,1,8,3,1,2,2,5,0,2,1,6,0,2

%N Decimal expansion of the greatest x such that 1/x + 1/(x+1) + 1/(x+3) = 2.

%C Equivalently, the least root of 2*x^3 + 5*x^2 - 2*x - 3;

%C Middle root: A316165;

%C Greatest root: A316166.

%C See A305328 for a guide to related sequences.

%F 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)))

%F ****

%F middle: -(5/6) - (1/12) sqrt(37) cos((1/3)(Pi - arctan((6 sqrt(1329))/53))) -

%F (1/12) sqrt(37) cos((1/3)(-Pi + arctan((6 sqrt(1329))/53))) +

%F (1/4) sqrt(37/3) sin((1/3)(Pi - arctan((6 sqrt(1329))/53))) -

%F (1/4) sqrt(37/3) sin((1/3)(-Pi + arctan((6 sqrt(1329))/53)))

%F ****

%F least: -(5/6) - (1/12) sqrt(37) cos((1/3)(Pi - arctan((6 sqrt(1329))/53))) -

%F (1/12) sqrt(37) cos(1/3(-Pi + arctan((6 sqrt(1329))/53))) -

%F (1/4) sqrt(37/3) sin((1/3)(Pi - arctan((6 sqrt(1329))/53))) +

%F (1/4) sqrt(37/3) sin(1/3)(-Pi + arctan((6 sqrt(1329))/53)))

%e greatest root: 0.83684889130097120054...

%e middle root: -0.67283324655316660799...

%e least root: -2.6640156447478045925...

%t a = 1; b = 1; c = 1; u = 0; v = 1; w = 3; d = 2;

%t r[x_] := a/(x + u) + b/(x + v) + c/(x + w);

%t t = Re[x /. ComplexExpand[Solve[r[x] == d, x]]]

%t N[t, 20]

%t u = N[t, 200];

%t u1 = RealDigits[u[[1]]]  (* A316166, greatest *)

%t u2 = RealDigits[u[[2]]]  (* A316164, least *)

%t u3 = RealDigits[u[[3]]]  (* A316165, middle *)

%t RealDigits[Root[1/x+1/(x+1)+1/(x+3)-2,3],10,120][[1]] (* _Harvey P. Dale_, Sep 06 2022 *)

%Y Cf. A305328, A316164, A316165.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Aug 08 2018