A360828 Decimal expansion of the ratio between the perimeter of the first Morley triangle of an isosceles right triangle and the perimeter of this isosceles right triangle.

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

Offset

0,2

Comments

The first Morley triangle, also called the Morley triangle, of any triangle is always equilateral (see Wikipedia link).

If an isosceles right triangle ABC has side lengths (a, a, a*sqrt(2)), then it has a circumradius R = a*sqrt(2)/2, and a perimeter P = (2 + sqrt(2))*a, and its first Morley triangle has side a' and perimeter P' = 3*a', with a' = 8*R*sin(Pi/6)*sin(Pi/12)*sin(Pi/12) = a*sqrt(2)*(2-sqrt(3))/2. This gives the ratio P'/P = (3/2) * (sqrt(2)-1) * (2-sqrt(3)) (see Illustration).

Links

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

Bernard Schott, Illustration of the Morley triangle of an isosceles right triangle.

Wikipedia, Morley's trisector theorem.

Formula

Equals (3/2) * (sqrt(2)-1) * (2-sqrt(3)).

Example

0.1664822842978339394003095728290656981743022858...

Maple

evalf((1/2)*(3*(sqrt(2)-1))*(2-sqrt(3)), 100);

Mathematica

RealDigits[(3/2)*(Sqrt[2] - 1)*(2 - Sqrt[3]), 10, 100][[1]] (* Amiram Eldar, Feb 28 2023 *)

Prog

(PARI) (3/2) * (sqrt(2)-1) * (2-sqrt(3)) \\ Michel Marcus, Mar 03 2023

Crossrefs

Cf. A359837 (ratio of perimeters in the case of an equilateral triangle), A360829 (ratio of areas in the case of an isosceles right triangle).

Sequence in context: A245297 A371504 A002421 * A209938 A165953 A045885

Adjacent sequences: A360825 A360826 A360827 * A360829 A360830 A360831

Keyword

nonn,cons

Author

Bernard Schott, Feb 28 2023

Status

approved