%I #21 Dec 31 2023 11:32:37

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

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

%U 2,8,5,7,7,0,7,8,9,0,4,7,4,9,9,4,2,5

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

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

%C Least root: A316131;

%C Middle root: A316132.

%C See A305328 for a guide to related sequences.

%F greatest root: -1/3 + (8/3)*cos((1/3)*arctan((9*sqrt(47))/17))

%F middle: -1/3 - (4/3)*cos((1/3)*arctan((9*sqrt(47))/17)) + (4*sin((1/3)*arctan((9*sqrt(47))/17)))/sqrt(3)

%F least: -1/3 - (4/3)*cos((1/3)*arctan((9*sqrt(47))/17)) - (4*sin((1/3)*arctan((9*sqrt(47))/17)))/sqrt(3)

%e greatest root: 2.0861301976514940912...

%e middle root: -0.57199326831620301856...

%e least root: -2.5141369293352910727...

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

%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]]] (* A316131 *)

%t RealDigits[u[[2]]] (* A316132 *)

%t RealDigits[u[[3]]] (* A316133 *)

%t RealDigits[Root[x^3+x^2-5x-3,3],10,120][[1]] (* _Harvey P. Dale_, Dec 31 2023 *)

%o (PARI) solve(x=2, 3, x^3+x^2-5*x-3) \\ _Altug Alkan_, Aug 01 2018

%Y Cf. A305328, A316131, A316132.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Jun 28 2018