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A316135
Decimal expansion of the middle x such that 1/x + 1/(x+2) + 1/(x+3) = 1 (negated).
4
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
OFFSET
1,2
COMMENTS
Equivalently, the middle root of x^3 + 2*x^2 - 4*x - 6;
Least root: A316134
Middle root: A316135;
Greatest root: A316136.
See A305328 for a guide to related sequences.
FORMULA
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]
EXAMPLE
greatest root: 1.8661982625090225055...
middle root: -1.2107558809591917224...
least root: -2.6554423815498307831...
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Jul 21 2018
STATUS
approved