A193211 Decimal expansion of the value of r that maximizes the Brahmagupta expression sqrt((-1+r+r^2+r^3)*(1-r+r^2+r^3)*(1+r-r^2+r^3)*(1+r+r^2-r^3))/4.

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

Offset

1,2

Comments

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 geometric progression 1:r:r^2:r^3 its common ratio r is limited to the range 1/t < r < t where t is the tribonacci constant (A058265). Consequently when r=1.6537455... it maximizes Brahmagupta's expression for the area of a convex cyclic quadrilateral whose sides form a geometric progression.

Links

Table of n, a(n) for n=1..103.

Wikipedia, Brahmagupta's formula.

Formula

r is the positive real root of the equation 1 + r^2 + 18*r^4 + 2*r^6 + 5*r^8 - 3*r^10 = 0. (Corrected by N. J. A. Sloane, Jan 14 2019. Thanks to Harvey P. Dale for pointing that the old expression was incorrect.)

Example

1.653745515077771929707906238366459714566223...

Maple

Digits:=200; fsolve( -3*r^10+5*r^8+2*r^6+18*r^4+r^2+1, r ); # N. J. A. Sloane, Jan 14 2019

Mathematica

RealDigits[r/.NMaximize[{Sqrt[(-1+r+r^2+r^3)(1-r+r^2+r^3)(1+r-r^2+r^3)(1+r+r^2-r^3)]/4, 3/5<r<9/5}, r, AccuracyGoal->120, PrecisionGoal->100, WorkingPrecision->240][[2]]][[1]]
RealDigits[r/.FindRoot[1+r^2+18r^4+2r^6+5r^8-3r^10==0, {r, 2}, WorkingPrecision -> 120]][[1]] (* Harvey P. Dale, Jan 14 2019 *)

Crossrefs

Sequence in context: A021157 A242494 A357107 * A195713 A306712 A109063

Adjacent sequences: A193208 A193209 A193210 * A193212 A193213 A193214

Keyword

nonn,cons

Author

Frank M Jackson, Sep 08 2011

Extensions

First Mathematica program fixed by Harvey P. Dale, Sep 10 2011

Second Mathematica program added by Harvey P. Dale, Jan 14 2019

Status

approved