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A316133
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Decimal expansion of the greatest x such that 1/x + 1/(x+1) + 1/(x+3) = 1.
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4
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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, 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, 2, 8, 5, 7, 7, 0, 7, 8, 9, 0, 4, 7, 4, 9, 9, 4, 2, 5
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OFFSET
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1,1
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COMMENTS
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Equivalently, the greatest root of x^3 + x^2 - 5*x - 3;
See A305328 for a guide to related sequences.
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LINKS
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FORMULA
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greatest root: -1/3 + (8/3)*cos((1/3)*arctan((9*sqrt(47))/17))
middle: -1/3 - (4/3)*cos((1/3)*arctan((9*sqrt(47))/17)) + (4*sin((1/3)*arctan((9*sqrt(47))/17)))/sqrt(3)
least: -1/3 - (4/3)*cos((1/3)*arctan((9*sqrt(47))/17)) - (4*sin((1/3)*arctan((9*sqrt(47))/17)))/sqrt(3)
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EXAMPLE
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greatest root: 2.0861301976514940912...
middle root: -0.57199326831620301856...
least root: -2.5141369293352910727...
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MATHEMATICA
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a = 1; b = 1; c = 1; u = 0; v = 1; w = 3; d = 1;
r[x_] := a/(x + u) + b/(x + v) + c/(x + w);
t = x /. ComplexExpand[Solve[r[x] == d, x]]
N[t, 20]
u = N[t, 200];
RealDigits[Root[x^3+x^2-5x-3, 3], 10, 120][[1]] (* Harvey P. Dale, Dec 31 2023 *)
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PROG
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(PARI) solve(x=2, 3, x^3+x^2-5*x-3) \\ Altug Alkan, Aug 01 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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