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Decimal expansion of the greatest x such that 1/x + 1/(x+3) + 1/(x+4) = 3.
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%I #15 Oct 07 2018 07:16:29

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

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

%U 9,5,1,5,4,0,0,0,7,9,5,4,8,3,0,7,4,5

%N Decimal expansion of the greatest x such that 1/x + 1/(x+3) + 1/(x+4) = 3.

%C Equivalently, the least root of 3*x^3 + 18*x^2 + 22 x - 12.

%C Least:: A316258

%C Middle: A316259;

%C See A305328 for a guide to related sequences.

%H Muniru A Asiru, <a href="/A316260/b316260.txt">Table of n, a(n) for n = 0..2000</a>

%F greatest root: -2 + (2/3)*sqrt(14)*cos((1/3)*arctan(sqrt(181/2)/9))

%F ****

%F middle: -2 - (1/3)*sqrt(14)*cos((1/3)*arctan(sqrt(181/2)/9)) + sqrt(14/3)*sin((1/3)*arctan(sqrt(181/2)/9))

%F ****

%F least: -2 - (1/3)*sqrt(14)*cos((1/3)*arctan(sqrt(181/2)/9)) - sqrt(14/3)*sin((1/3)*arctan(sqrt(181/2)/9))

%e greatest root: 0.4033761543003640184...

%e middle root: -2.623324901998131634... [Corrected by _A.H.M. Smeets_, Sep 17 2018]

%e least root: -3.780051252302232384...

%p evalf(solve(3*x^3+18*x^2+22*x-12=0,x)[1],90); # _Muniru A Asiru_, Oct 07 2018

%t a = 1; b = 1; c = 1; u = 0; v = 3; w = 4; 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 y = Re[N[t, 200]];

%t RealDigits[y[[1]]] (* A316259, middle *)

%t RealDigits[y[[2]]] (* A316258, least *)

%t RealDigits[y[[3]]] (* A316260, greatest *)

%Y Cf. A305328, A316258, A316259.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Sep 14 2018