A199861 The decimal expansion (unsigned) of the value of d that maximizes the Brahmagupta expression given below.

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, 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, 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

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

Table of n, a(n) for n=0..102.

Wikipedia, Brahmagupta's formula.

Index entries for algebraic numbers, degree 12

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

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

Sequence in context: A164325 A198880 A201315 * A190256 A013671 A019807

Adjacent sequences: A199858 A199859 A199860 * A199862 A199863 A199864

Keyword

nonn,cons

Author

Frank M Jackson, Nov 11 2011

Status

approved