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URL: https://oeis.org/A316131

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A316131
Decimal expansion of the least x such that 1/x + 1/(x+1) + 1/(x+3) = 1, negated.
4
2, 5, 1, 4, 1, 3, 6, 9, 2, 9, 3, 3, 5, 2, 9, 1, 0, 7, 2, 6, 9, 3, 7, 7, 4, 8, 6, 6, 9, 6, 2, 2, 1, 7, 4, 7, 8, 0, 5, 2, 4, 7, 6, 3, 0, 0, 7, 4, 5, 4, 0, 4, 5, 9, 2, 2, 2, 1, 6, 7, 1, 3, 9, 4, 2, 0, 9, 3, 4, 1, 6, 5, 7, 2, 9, 1, 7, 7, 3, 5, 9, 0, 7, 5, 8, 0
OFFSET
1,1
COMMENTS
Equivalently, the least root of x^3 + x^2 - 5*x - 3;
Middle root: A316132;
Greatest root: A316133.
See A305328 for a guide to related sequences.
FORMULA
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)
EXAMPLE
greatest root: 2.0861301976514940912...
middle root: -0.57199326831620301856...
least root: -2.5141369293352910727...
MATHEMATICA
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[u[[1]]] (* A316131 *)
RealDigits[u[[2]]] (* A316132 *)
RealDigits[u[[3]]] (* A316133 *)
RealDigits[-x/.FindRoot[1/x+1/(x+1)+1/(x+3)==1, {x, -2.5}, WorkingPrecision -> 120]][[1]] (* Harvey P. Dale, Jun 27 2021 *)
PROG
(PARI) solve(x=-3, -2, x^3+x^2-5*x-3) \\ Jianing Song, Aug 01 2018
(PARI) polrootsreal(x^3 - x^2 - 5*x + 3)[3] \\ Charles R Greathouse IV, Apr 10 2026
CROSSREFS
KEYWORD
nonn,cons,changed
AUTHOR
Clark Kimberling, Jun 26 2018
STATUS
approved