This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A342977 #42 Feb 28 2024 07:03:16
%S A342977 2,8,5,3,9,8,1,6,3,3,9,7,4,4,8,3,0,9,6,1,5,6,6,0,8,4,5,8,1,9,8,7,5,7,
%T A342977 2,1,0,4,9,2,9,2,3,4,9,8,4,3,7,7,6,4,5,5,2,4,3,7,3,6,1,4,8,0,7,6,9,5,
%U A342977 4,1,0,1,5,7,1,5,5,2,2,4,9,6,5,7,0,0,8,7,0,6,3,3,5,5,2,9,2
%N A342977 Decimal expansion of (Pi - 2) / 4.
%C A342977 The constant represents the area of a circular segment bounded by an arc of Pi/2 radians (the right angle) of a unit circle and by a chord of the length of sqrt(2). Four such segments result when a square with the side length of sqrt(2) is circumscribed by a unit circle. The area of each segment is:
%C A342977 A = (R^2 / 2) * (theta - sin(theta))
%C A342977 A = (1^2 / 2) * (Pi/2 - sin(Pi/2))
%C A342977 A = (1 / 2) * (Pi/2 - 1)
%C A342977 A = (Pi - 2) / 4 = 0.28539816...
%C A342977 where Pi = 3.14159265... (A000796) is the area bounded by the unit circle, and 2 is the area of the inscribed square.
%C A342977 Apart from the first digit this is the same as Pi/4 = 0.78539816... (A003881), the area of a circular sector bounded by the arc of Pi/2 = 1.57079632... (A019669) radians of the unit circle and by two radii of unit length, and 1/2 = 0.5 (A020761) is one-quarter of the area of the inscribed square.
%C A342977 The constant is close to 2/7 = 0.28571428... (2 * A020806) and Pi/11 = 0.28559933... (A019678). The equation (x - 2)/4 = x/11 has a solution x = 22/7 = 3.14285714... (A068028), which is an approximation of Pi.
%C A342977 The best rational approximation of the constant using small positive integers (less than 1000) is 129/452 = 0.28539823..., the next best approximation is 4771/16717 = 0.28539809...
%C A342977 The reciprocal of the constant is:
%C A342977 1/A = 4 / (Pi - 2) = 3.50387678... (A309091).
%C A342977 The sagitta (height) of the circular segment is:
%C A342977 h = R * (1 - cos(theta/2))
%C A342977 h = 1 * (1 - cos(Pi/4))
%C A342977 h = 1 - sqrt(2) / 2
%C A342977 h = 1 - 1 / sqrt(2) = 0.29289321... (A268682).
%H A342977 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A342977 Equals Integral_{x=-sqrt(2)/2..sqrt(2)/2} Integral_{y=sqrt(2)/2..sqrt(1-x^2)} dy dx.
%F A342977 Equals Sum_{k>=1} (-1)^(k + 1)/(4*k^2 - 1). - _Amiram Eldar_, Jun 08 2021
%F A342977 Continued fraction: 1/(3 + 3/(4 + 15/(4 + 35/(4 + ... + (4*n^2 - 1)/(4 + ...). - _Peter Bala_, Feb 22 2024
%e A342977 0.2853981633974483...
%t A342977 RealDigits[Pi/4 - 1/2, 10, 100][[1]] (* _Amiram Eldar_, Jun 08 2021 *)
%o A342977 (PARI) (Pi - 2) / 4
%Y A342977 Cf. A000796, A019669, A019678, A020761, A020806, A068028, A268682, A309091. Essentially the same as A003881.
%K A342977 nonn,cons
%O A342977 0,1
%A A342977 _Michal Paulovic_, Apr 01 2021