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A199589
Decimal expansion of the greatest root of 6x^3 - 6x - 2 = 0.
2
1, 1, 3, 7, 1, 5, 8, 0, 4, 2, 6, 0, 3, 2, 5, 7, 6, 1, 2, 8, 3, 7, 6, 6, 7, 9, 5, 1, 9, 2, 0, 0, 9, 8, 7, 6, 2, 5, 8, 1, 3, 6, 0, 3, 9, 4, 2, 2, 9, 9, 0, 6, 5, 8, 5, 9, 6, 2, 8, 8, 7, 9, 6, 4, 9, 4, 4, 2, 5, 1, 0, 6, 6, 5, 6, 8, 5, 0, 9, 4, 5, 4, 9, 8, 5, 3, 1, 6, 7, 7, 7, 6, 7, 8, 9, 9, 7, 7, 0
OFFSET
1,3
COMMENTS
If the side lengths of a quadrilateral form a harmonic progression in the ratio 1 : 1/(1+d) : 1/(1+2d) : 1/(1+3d) where d is the common difference between the denominators of the harmonic progression, then the triangle inequality condition requires that d be in the range f < d < g, where g = 1.1371580... and is the greatest root of the equation: 2 + 6d - 6d^3 = 0. The value of f is given in A199590.
FORMULA
sqrt(4/3)*cos(Pi/18). - Charles R Greathouse IV, Nov 10 2011
EXAMPLE
1.13715804260325761283766795192009876258136039422990658596288796494425...
MATHEMATICA
N[Reduce[2+6d-6d^3==0, d], 100]
PROG
(PARI) real(polroots(6*x^3-6*x-2)[3]) \\ Charles R Greathouse IV, Nov 10 2011
(PARI) polrootsreal(6*x^3-6*x-2)[3] \\ Charles R Greathouse IV, Apr 14 2014
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Frank M Jackson, Nov 08 2011
STATUS
approved