cp's OEIS Frontend

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

%I A359837 #30 Feb 05 2025 10:03:36
%S A359837 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 A359837 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 A359837 9,1,3,3,4,3,5,7,1,9,9,7,6,2,1,3,4,2,5,3,2,7
%N A359837 Decimal expansion of the unsigned ratio of similitude between an equilateral reference triangle and its first Morley triangle.
%C A359837 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 A359837 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 A359837 Wikipedia, <a href="https://en.wikipedia.org/wiki/Morley%27s_trisector_theorem"> Morley's trisector theorem</a>.
%H A359837 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>.
%F A359837 Equals sin(Pi/18)/cos(Pi/9).
%F A359837 A root of x^3+3*x^2-6*x+1.
%F A359837 Equals A019819/A019879. - _Michel Marcus_, Jan 15 2023
%F A359837 Equals 8 * A020760 * A019829^3. - _Bernard Schott_, Feb 06 2023
%e A359837 0.1847925309040953727013520475722037558709135597926517172356...
%t A359837 RealDigits[Sin[Pi/18]/Cos[Pi/9], 10, 100][[1]]
%t A359837 N[Solve[x^3 + 3*x^2 - 6*x + 1 == 0, {x}][[2]], 90]
%o A359837 (PARI) sin(Pi/18)/cos(Pi/9) \\ _Michel Marcus_, Jan 15 2023
%Y A359837 Cf. A019819, A019829, A019879, A020760.
%K A359837 easy,nonn,cons
%O A359837 0,2
%A A359837 _Frank M Jackson_, Jan 14 2023