A316246 - OEIS

%I #13 May 28 2023 13:43:53

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

%T 5,8,5,7,5,9,8,7,3,8,2,6,2,9,1,3,2,2,0,1,9,9,0,1,2,7,7,0,6,3,9,2,5,8,

%U 4,5,3,7,8,4,1,6,1,0,5,3,4,8,6,3,2,1

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

%C Equivalently, the least root of 3*x^3 + 6*x^2 - 2;

%C Middle root: A316247;

%C Greatest root: A316248.

%C See A305328 for a guide to related sequences.

%F greatest root: -2/3 + (4/3)*cos((1/3)*arctan(3*sqrt(7)))

%F ****

%F middle: -2/3 - (2/3)*cos((1/3)*arctan(3*sqrt(7))) + 2*sin((1/3)*arctan(3*sqrt(7)))/sqrt(3)

%F ****

%F least: -2/3 - (2/3)*cos((1/3)*arctan(3*sqrt(7))) - 2*sin((1/3)*arctan(3*sqrt(7)))/sqrt(3)

%e greatest root: 0.5148689384387165869...

%e middle root: -0.7223517244643762951...

%e least root: -1.792517213974340291...

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

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

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

%t N[t, 20]

%t u = N[t, 200];

%t RealDigits[u[[1]]]  (* A316246, greatest *)

%t RealDigits[u[[2]]]  (* A316247, least *)

%t RealDigits[u[[3]]]  (* A316248, middle *)

%t RealDigits[Root[1/x+1/(x+1)+1/(x+2)-3,1],10,120][[1]] (* _Harvey P. Dale_, May 28 2023 *)

%Y Cf. A305328, A316247, A316248.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Aug 19 2018

%E a(86) corrected by _Andrew Howroyd_, Nov 04 2018