%I #34 Jul 16 2024 08:08:59

%S 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,

%T 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,

%U 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

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

%C 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).

%H G. C. Greubel, <a href="/A193355/b193355.txt">Table of n, a(n) for n = 0..10000</a>

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

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

%F 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

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

%e 0.6506451422...

%p evalf(Pi/(2+2*sqrt(2)),120); # _Muniru A Asiru_, Sep 30 2018

%t N[Pi/(2 + 2*Sqrt[2]), 100]

%t Realdigits[Pi/(2 + 2*Sqrt[2]), 10, 100][[1]] (* _G. C. Greubel_, Sep 29 2018 *)

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

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)/(2 + 2*Sqrt(2)); // _G. C. Greubel_, Sep 29 2018

%K easy,nonn,cons

%O 0,1

%A _Frank M Jackson_, Jul 24 2011