

A193211


The decimal expansion of the value of r that maximizes the Brahmagupta expression Sqrt((1+r+r^2+r^3)(1r+r^2+r^3)(1+rr^2+r^3)(1+r+r^2r^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
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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 equation 1+r^2+18*r^4+2r^6+5r^83r^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)(1r+r^2+r^3)(1+rr^2+r^3)(1+r+r^2r^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^83r^10==0, {r, 2}, WorkingPrecision > 120]][[1]] (* Harvey P. Dale, Jan 14 2019 *)


CROSSREFS

Sequence in context: A143304 A021157 A242494 * 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



