%I #17 Oct 27 2023 12:06:28

%S 2,2,7,1,0,6,4,4,8,2,9,4,3,8,1,2,0,3,0,1,1,1,4,3,3,5,2,5,3,2,3,4,4,6,

%T 1,8,3,7,7,5,4,0,5,3,1,2,9,8,6,7,4,9,6,2,9,3,2,5,4,0,3,5,4,5,5,0,4,8,

%U 1,2,6,1,0,0,0,1,6,0,1,8,4,3,7,1,1,6,7,7,4,5,2,8,4,9,4,9,4,5,8,6,3,5,8

%N The decimal expansion (unsigned) of the value of d that maximizes the Brahmagupta expression given below.

%C 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.

%C 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.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Brahmagupta%27s_formula">Brahmagupta's formula</a>.

%H <a href="/index/Al#algebraic_12">Index entries for algebraic numbers, degree 12</a>

%F 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.

%e -0.22710644829438120301114335253234461837754...

%t 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]]

%o (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

%o (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

%Y Cf. A010503, A193180, A193211, A199589, A199590.

%K nonn,cons

%O 0,1

%A _Frank M Jackson_, Nov 11 2011