

A199861


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


0



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
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OFFSET

0,1


COMMENTS

Brahmagupta expression sqrt((1+1/(1+d)+1/(1+2d)+1/(1+3d)) * (11/(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.


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))(11/(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


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



