OFFSET
0,1
COMMENTS
Brahmagupta expression sqrt((-1+1/(1+d)+1/(1+2d)+1/(1+3d)) * (1-1/(1+d)+1/(1+2d)+1/(1+3d)) * (1+1/(1+d)-1/(1+2d)+1/(1+3d)) * (1+1/(1+d)+1/(1+2d)-1/(1+3d)))/4 for d in the interval [-1/3, inf] where 1/(1+d), 1/(1+2d) and 1/(1+3d) are always positive.
The area of a convex quadrilateral with fixed sides is maximal when it is organized as a convex cyclic quadrilateral. Furthermore in order that a quadrilateral can have sides in a harmonic progression 1 : 1/(1+d) : 1/(1+2d) : 1/(1+3d) its denominator's common difference d is limited to the range f < d < g where f is the constant A199590 and g is the constant A199589. Consequently when d=-0.2271064482... it maximizes Brahmagupta's expression for the area of a convex cyclic quadrilateral whose sides form a harmonic progression.
LINKS
FORMULA
d is the largest real root of the equation 1323d^12 + 9711d^11 + 32535d^10 + 67005d^9 + 94338d^8 + 94761d^7 + 68955d^6 + 36367d^5 + 13740d^4 + 3619d^3 + 630d^2 + 65d + 3 = 0.
EXAMPLE
-0.22710644829438120301114335253234461837754...
MATHEMATICA
RealDigits[d/.NMaximize[{Sqrt[(-1+1/(1+d)+1/(1+2d)+1/(1+3d))(1-1/(1+d)+1/(1+2d)+1/(1+3d))(1+1/(1+d)-1/(1+2d)+1/(1+3d))(1+1/(1+d)+1/(1+2d)-1/(1+3d))]/4, -1/4<d<9/8}, d, AccuracyGoal->120, PrecisionGoal->100, WorkingPrecision->240][[2]]][[1]]
PROG
(PARI) real(polroots(1323*d^12 + 9711*d^11 + 32535*d^10 + 67005*d^9 + 94338*d^8 + 94761*d^7 + 68955*d^6 + 36367*d^5 + 13740*d^4 + 3619*d^3 + 630*d^2 + 65*d + 3)[4]) \\ Charles R Greathouse IV, Nov 11 2011
(PARI) polrootsreal(1323*x^12 - 9711*x^11 + 32535*x^10 - 67005*x^9 + 94338*x^8 - 94761*x^7 + 68955*x^6 - 36367*x^5 + 13740*x^4 - 3619*x^3 + 630*x^2 - 65*x + 3)[1] \\ Charles R Greathouse IV, Oct 27 2023
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Frank M Jackson, Nov 11 2011
STATUS
approved