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A316135
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Decimal expansion of the middle x such that 1/x + 1/(x+2) + 1/(x+3) = 1 (negated).
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4
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1, 2, 1, 0, 7, 5, 5, 8, 8, 0, 9, 5, 9, 1, 9, 1, 7, 2, 2, 3, 8, 0, 2, 1, 4, 5, 6, 7, 4, 4, 8, 3, 1, 4, 3, 3, 2, 9, 2, 7, 4, 3, 9, 1, 9, 9, 1, 5, 5, 1, 8, 8, 3, 5, 3, 5, 9, 4, 5, 3, 7, 2, 1, 4, 6, 0, 8, 5, 2, 1, 2, 6, 9, 2, 1, 5, 6, 6, 9, 6, 0, 8, 3, 3, 7, 5
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OFFSET
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1,2
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COMMENTS
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Equivalently, the middle root of x^3 + 2*x^2 - 4*x - 6;
See A305328 for a guide to related sequences.
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LINKS
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FORMULA
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greatest root: -(2/3) + 8/3 cos[1/3 arctan[(3 sqrt[303])/37]]
middle: -(2/3) - 4/3 cos[1/3 arctan[(3 sqrt[303])/37]] + (4 sin[1/3 arctan[(3 sqrt[303])/37]])/sqrt[3]
least: -(2/3) - 4/3 cos[1/3 arctan[(3 sqrt[303])/37]] - (4 sin[1/3 arctan[(3 sqrt[303])/37]])/sqrt[3]
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EXAMPLE
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greatest root: 1.8661982625090225055...
middle root: -1.2107558809591917224...
least root: -2.6554423815498307831...
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MATHEMATICA
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a = 1; b = 1; c = 1; u = 0; v = 2; 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[u[[1]]] (* A316134, least *)
RealDigits[u[[2]]] (* A316135, middle *)
RealDigits[u[[3]]] (* A316136, greatest *)
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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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