A359837 - OEIS

%I #28 Feb 18 2023 15:29:05

%S 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,

%T 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,

%U 9,1,3,3,4,3,5,7,1,9,9,7,6,2,1,3,4,2,5,3,2,7

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

%C 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.

%C 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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Morley%27s_trisector_theorem"> Morley's trisector theorem</a>.

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

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

%F Equals A019819/A019879. - _Michel Marcus_, Jan 15 2023

%F Equals 8 * A020760 * A019829^3. - _Bernard Schott_, Feb 06 2023

%e 0.1847925309040953727013520475722037558709135597926517172356...

%t RealDigits[Sin[Pi/18]/Cos[Pi/9], 10, 100][[1]]

%t N[Solve[x^3 + 3*x^2 - 6*x + 1 == 0, {x}][[2]], 90]

%o (PARI) sin(Pi/18)/cos(Pi/9) \\ _Michel Marcus_, Jan 15 2023

%Y Cf. A019819, A019829, A019879, A020760.

%K easy,nonn,cons

%O 0,2

%A _Frank M Jackson_, Jan 14 2023