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A193180
Decimal expansion of the value of d that maximizes the expression (1+1/(1-d)+1/(1-2d))*(-1+1/(1-d)+1/(1-2d))*(1-1/(1-d)+1/(1-2d))*(1+1/(1-d)-1/(1-2d)) for d in the interval [-oo, 1/2] where 1/(1-d) and 1/(1-2d) are always positive.
1
2, 1, 5, 8, 5, 6, 1, 2, 2, 6, 1, 1, 5, 1, 8, 2, 0, 1, 1, 8, 1, 4, 9, 7, 1, 0, 5, 7, 1, 6, 2, 7, 8, 3, 0, 3, 2, 9, 7, 7, 4, 0, 9, 9, 4, 8, 6, 4, 4, 6, 5, 1, 7, 2, 4, 4, 4, 0, 3, 0, 9, 4, 4, 9, 5, 8, 7, 1, 6, 9, 0, 4, 3, 0, 1, 6, 5, 7, 3, 5, 8, 7, 8, 5, 3, 0, 5, 5, 7, 5, 3, 8, 7, 2, 4, 9, 5, 9, 1, 1
OFFSET
0,1
COMMENTS
If the sides of a triangle form a harmonic progression in the ratio 1 : 1/(1-d) : 1/(1-2d) then when d = 0.2158561226... it uniquely maximizes the area of the triangle. This triangle has approximate internal angles of 33.956, 45.423 and 100.621 degrees.
EXAMPLE
0.2158561226...
MATHEMATICA
NMaximize[{Sqrt[(1+1/(1-d)+1/(1-2d))(-1+1/(1-d)+1/(1-2d))(1-1/(1-d)+1/(1-2d))(1+1/(1-d)-1/(1-2d))]/4, 0<d<(25/100)}, d, AccuracyGoal->120, PrecisionGoal->100, WorkingPrecision->240]
CROSSREFS
Sequence in context: A257574 A316293 A377363 * A201743 A167816 A316292
KEYWORD
nonn,cons,easy
AUTHOR
Frank M Jackson, Jul 17 2011
STATUS
approved