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A193211 The 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
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 (list; constant; graph; refs; listen; history; text; internal format)
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 positive real root of equation 1+r^2+2r^6+5r^8-3r^10=0

EXAMPLE

1.653745515077771929707906238366459714566223...

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]]

CROSSREFS

Sequence in context: A143304 A021157 A242494 * A195713 A109063 A110390

Adjacent sequences:  A193208 A193209 A193210 * A193212 A193213 A193214

KEYWORD

nonn,cons

AUTHOR

Frank M Jackson, Sep 08 2011

EXTENSIONS

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

STATUS

approved

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Last modified December 8 12:55 EST 2016. Contains 278945 sequences.