OFFSET
0,1
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 f = -0.257772801... and is the greatest root of the equation: 2 + 12d + 18d^2 + 6d^3 = 0. The value of g is given in A199589.
FORMULA
sqrt(4/3)*sin(Pi*2/9) - 1. - Charles R Greathouse IV, Nov 10 2011
EXAMPLE
-0.257772801031440844729449397270635822708944125700977319782314639395808...
MATHEMATICA
N[Reduce[2+12d+18d^2+6d^3==0, d], 100]
PROG
(PARI) real(polroots(6*x^3+18*x^2+12*x+2)[3]) \\ Charles R Greathouse IV, Nov 10 2011
(PARI) polrootsreal(6*x^3-18*x^2+12*x-2)[1] \\ Charles R Greathouse IV, Oct 27 2023
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Frank M Jackson, Nov 08 2011
EXTENSIONS
a(99) corrected by Sean A. Irvine, Jul 25 2021
STATUS
approved