A359837 Decimal expansion of the unsigned ratio of similitude between an equilateral reference triangle and its first Morley triangle.

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

Offset

0,2

Comments

The first Morley triangle of any reference triangle is always equilateral. Therefore a reference equilateral triangle and its first Morley triangle will be in a homothetic relationship. This sequence is the real terms of a constant that is the ratio of similitude of the homothety. The constant is negative.

If an equilateral triangle has a side a, a circumradius R and a first Morley triangle with side a', then a = R*sqrt(3) and a' = 8*R*(sin(Pi/9))^3, so the ratio of similitude between the two triangles is a'/a = (8/sqrt(3)) * (sin(Pi/9))^3. - Bernard Schott, Feb 06 2023

Links

Table of n, a(n) for n=0..89.

Wikipedia, Morley's trisector theorem.

Formula

Equals sin(Pi/18)/cos(Pi/9).

A root of x^3+3*x^2-6*x+1.

Equals A019819/A019879. - Michel Marcus, Jan 15 2023

Equals 8 * A020760 * A019829^3. - Bernard Schott, Feb 06 2023

Example

0.1847925309040953727013520475722037558709135597926517172356...

Mathematica

RealDigits[Sin[Pi/18]/Cos[Pi/9], 10, 100][[1]]
N[Solve[x^3 + 3*x^2 - 6*x + 1 == 0, {x}][[2]], 90]

Prog

(PARI) sin(Pi/18)/cos(Pi/9) \\ Michel Marcus, Jan 15 2023

Crossrefs

Cf. A019819, A019829, A019879, A020760.

Sequence in context: A254246 A369103 A254375 * A019684 A244210 A019867

Adjacent sequences: A359834 A359835 A359836 * A359838 A359839 A359840

Keyword

easy,nonn,cons

Author

Frank M Jackson, Jan 14 2023

Status

approved