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