A193355 Decimal expansion of Pi/(2 + 2*sqrt(2)).

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

Offset: 0

Author

Frank M Jackson, Jul 24 2011

Keywords

• easy
• nonn
• cons

Comments

This is the first of the three angles (in radians) of a unique triangle that is right angled and where the angles are in a harmonic progression: Pi/(2+2*sqrt(2)) (this sequence), Pi/(2+sqrt(2)) (A193373), Pi/2 (A019669). The angles (in degrees) are approximately 37.279, 52.721, 90. The common difference between the denominators of the harmonic progression is sqrt(2).

Examples

			0.6506451422...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Index entries for transcendental numbers

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)/(2 + 2*Sqrt(2)); // G. C. Greubel, Sep 29 2018

Maple

evalf(Pi/(2+2*sqrt(2)),120); # Muniru A Asiru, Sep 30 2018

Mathematica

N[Pi/(2 + 2*Sqrt[2]), 100]
RealDigits[Pi/(2 + 2*Sqrt[2]), 10, 100][[1]] (* G. C. Greubel, Sep 29 2018 *)

PARI

default(realprecision,100); Pi/(2+2*sqrt(2))

Formula

Equals Pi/(2+2*sqrt(2)).

Equals Integral_{x=0..Pi/2} cos(x)^2/(1 + sin(x)^2) dx = Integral_{x=0..Pi/2} sin(x)^2/(1 + cos(x)^2) dx. - Amiram Eldar, Aug 16 2020

Equals 4*Sum_{k >= 0} (-1)^k/((4*k + 1)*(4*k + 2)*(4*k + 3)). - Peter Bala, Jul 15 2024

Equals Integral_{x=0..1} sqrt(1 - x^2)/(1 + x^2) dx. - Kritsada Moomuang, Jun 05 2025

Equals A247719 - A019669 = A000796*A268683. - R. J. Mathar, Jul 22 2025