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A316250
Decimal expansion of the middle x such that 1/x + 1/(x+1) + 1/(x+3) = 3.
4
7, 4, 3, 6, 8, 5, 3, 1, 4, 3, 2, 2, 9, 1, 6, 3, 7, 3, 3, 9, 5, 0, 6, 8, 3, 9, 4, 3, 3, 9, 2, 1, 5, 2, 8, 3, 4, 7, 9, 6, 4, 1, 8, 1, 0, 4, 8, 8, 3, 5, 4, 5, 1, 0, 8, 7, 2, 3, 2, 9, 4, 3, 8, 8, 9, 0, 3, 8, 8, 9, 2, 8, 2, 4, 7, 2, 3, 5, 7, 6, 1, 2, 9, 1, 5, 8
OFFSET
0,1
COMMENTS
Equivalently, the middle root of 3*x^3 + 9*x^2 + x - 3;
Least root: A316249;
Greatest root: A316251.
See A305328 for a guide to related sequences.
FORMULA
greatest root: -1 + (2/3)*sqrt(2)*cos(Pi/3 - (1/3)*arctan(sqrt(431)/9))) + (2/3)*sqrt(2)*cos(-Pi/3 + (1/3)*arctan(sqrt(431)/9)))
****
middle: -1 - (1/3)*sqrt(2)*cos(Pi/3 - (1/3)*arctan(sqrt(431)/9))) - (1/3)*sqrt(2)*cos(-Pi/3 + (1/3)*arctan(sqrt(431)/9))) + sqrt(2/3)*sin(Pi/3 - (1/3)*arctan(sqrt(431)/9))) - sqrt(2/3)*sin(-Pi/3 + (1/3)*arctan(sqrt(431)/9)))
****
least: -1 - (1/3)*sqrt(2)*cos(Pi/3 - (1/3)*arctan(sqrt(431)/9))) - (1/3)*sqrt(2)*cos(-Pi/3 + (1/3)*arctan(sqrt(431)/9))) - sqrt(2/3)*sin(Pi/3 - (1/3)*arctan(sqrt(431)/9))) + sqrt(2/3)*sin(-Pi/3 + (1/3)*arctan(sqrt(431)/9)))
EXAMPLE
greatest root: 0.4896787901169438959...
middle root: -0.7436853143229163734...
least root: -2.745993475794027522...
MATHEMATICA
a = 1; b = 1; c = 1; u = 0; v = 1; w = 3; d = 3;
r[x_] := a/(x + u) + b/(x + v) + c/(x + w);
t = x /. ComplexExpand[Solve[r[x] == d, x]]
N[t, 20]
y = Re[N[t, 200]];
RealDigits[y[[1]]] (* A316251, greatest *)
RealDigits[y[[2]]] (* A316249, least *)
RealDigits[y[[3]]] (* A316250, middle *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Aug 21 2018
STATUS
approved