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.

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

Author

Frank M Jackson, Jul 17 2011

Keywords

• nonn
• cons
• easy

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.

Examples

			0.2158561226...
		

Programs

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, 0120, PrecisionGoal->100, WorkingPrecision->240]