A193180 The 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 [-inf, 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

0,1

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.

Links

Table of n, a(n) for n=0..99.

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: A109233 A257574 A316293 * A201743 A167816 A316292

Adjacent sequences: A193177 A193178 A193179 * A193181 A193182 A193183

Keyword

nonn,cons,easy

Author

Frank M Jackson, Jul 17 2011

Status

approved