A358981 - OEIS

%I #16 Mar 08 2024 11:28:07

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

%T 3,6,3,3,0,0,2,1,8,1,9,6,7,2,4,4,0,1,1,6,6,4,4,3,6,3,1,1,9,2,3,9,6,2,

%U 2,2,1,4,5,3,4,8,6,9,6,5,6,9,3,9,0,5,8,3,9,5,0,9,1,3,9,3,5,4,5,4

%N Decimal expansion of Pi/3 - sqrt(3)/4.

%C The constant is the area of a circular segment bounded by an arc of 2*Pi/3 radians (120 degrees) of a unit circle and by a chord of length sqrt(3). Three such segments result when an equilateral triangle with side length sqrt(3) is circumscribed by a unit circle. The area of each segment is:

%C   A = (R^2 / 2) * (theta - sin(theta))

%C   A = (1^2 / 2) * (2*Pi/3 - sin(2*Pi/3))

%C   A = (1 / 2) * (2*Pi/3 - sqrt(3)/2)

%C   A = Pi/3 - sqrt(3)/4 = (Pi - 3*sqrt(3)/4) / 3 = 0.61418484...

%C where Pi (A000796) is the area of the circle, and 3*sqrt(3)/4 (A104954) is the area of the inscribed equilateral triangle.

%C The sagitta (height) of the circular segment is:

%C   h = R * (1 - cos(theta/2))

%C   h = 1 * (1 - cos(Pi/3))

%C   h = 1 - 1/2 = 0.5 (A020761)

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals A019670 - A120011. - _Omar E. Pol_, Dec 08 2022

%F Equals A093731 / 2. - _Michal Paulovic_, Mar 08 2024

%e 0.6141848493043784...

%p evalf(Pi/3-sqrt(3)/4);

%t RealDigits[Pi/3 - Sqrt[3]/4, 10, 100][[1]]

%o (PARI) Pi/3 - sqrt(3)/4

%Y Cf. A000796, A019670, A020761, A093731, A104954, A120011.

%K nonn,cons

%O 0,1

%A _Michal Paulovic_, Dec 08 2022