A176269 Decimal expansion of the value of d that maximizes the area of a triangle whose angles form a harmonic progression in the ratio 1 : 1/(1-d) : 1/(1-2d), whose side length opposite the largest or smallest angle remains constant and for d in the interval [-oo, 1/2] where 1/(1-d) and 1/(1-2d) are always positive.

3, 7, 7, 8, 3, 2, 3, 0, 1, 3, 3, 3, 7, 3, 5, 1, 5, 9, 9, 4, 0, 9, 9, 7, 3, 8, 4, 0, 0, 7, 3, 9, 6, 2, 2, 4, 6, 0, 6, 2, 2, 9, 2, 2, 7, 1, 0, 3, 1, 3, 6, 7, 4, 6, 2, 1, 7, 0, 4, 4, 8, 0, 6, 0, 8, 7, 6, 6, 6, 5, 3, 7, 1, 7, 0, 4, 8, 3, 3, 5, 5, 3, 6, 4, 9, 0, 5, 8, 4, 4, 7, 0, 5, 6, 7, 2, 7, 3, 1, 3, 5, 7

Offset

0,1

Comments

If the angles of a triangle form a harmonic progression in the ratio 1 : 1/(1-d) : 1/(1-2d) where d is the common difference between the denominators of the harmonic progression, then the angles are expressed as

Pi*(1-d)*(1-2d)/(2d^2-6d+3), Pi*(1-2d)/(2d^2-6d+3), Pi*(1-d)/(2d^2-6d+3).

When d = 0.3778323013... it uniquely maximizes the area of this triangle for a constant side length opposite the largest or smallest angle. This triangle has approximate internal angles 26.866 degrees, 43.181 degrees, 109.953 degrees.

Links

Table of n, a(n) for n=0..101.

Example

0.3778323013337351599409973840073962246062292271031367462170448060876665371704...

Mathematica

NMaximize[{Sqrt[(Sin[Pi*(1-d)*(1-2d)/(2d^2-6d+3)]+Sin[Pi*(1-2d)/(2d^2-6d+3)]+Sin[Pi*(1-d)/(2d^2-6d+3)])*(-Sin[Pi*(1-d)*(1-2d)/(2d^2-6d+3)]+Sin[Pi*(1-2d)/(2d^2-6d+3)]+Sin[Pi*(1-d)/(2d^2-6d+3)])*(Sin[Pi*(1-d)*(1-2d)/(2d^2-6d+3)]-Sin[Pi*(1-2d)/(2d^2-6d+3)]+Sin[Pi*(1-d)/(2d^2-6d+3)])*(Sin[Pi*(1-d)*(1-2d)/(2d^2-6d+3)]+Sin[Pi*(1-2d)/(2d^2-6d+3)]-Sin[Pi*(1-d)/(2d^2-6d+3)])]/(4*Sin[Pi*(1-d)(1-2d)/(2d^2-6d+3)]^2), d<1/2}, d, AccuracyGoal->120, PrecisionGoal->100, WorkingPrecision->240]

Crossrefs

Sequence in context: A349604 A325894 A370517 * A393776 A343644 A081522

Adjacent sequences: A176266 A176267 A176268 * A176270 A176271 A176272

Keyword

nonn,cons

Author

Frank M Jackson, Aug 04 2011

Status

approved