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 A224268 #34 Feb 07 2025 13:17:34
%S A224268 9,2,7,0,3,7,3,3,8,6,5,0,6,8,5,9,5,9,2,1,6,9,2,5,1,7,3,5,9,7,6,3,0,0,
%T A224268 2,3,1,0,8,7,9,9,4,1,1,7,6,0,8,8,3,4,5,2,7,9,2,9,6,4,0,2,2,5,2,8,0,1,
%U A224268 0,8,8,8,4,1,9,0,5,9,9,8,9,1,7,8,6,3,5
%N A224268 Decimal expansion of Product_{n>=1} (1 - 1/(4n+1)^2).
%D A224268 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.
%H A224268 Vincenzo Librandi, <a href="/A224268/b224268.txt">Table of n, a(n) for n = 0..1000</a>
%H A224268 Peter Bala, <a href="/A096427/a096427.pdf">Notes on the constants A096427 and A224268</a>.
%F A224268 Equals Gamma(1/4)^2/(8*sqrt(Pi)) = L/(4*sqrt(2)), where L is the Lemniscate constant (A064853).
%F A224268 From _Peter Bala_, Feb 26 2019: (Start)
%F A224268 C = (Pi/4)*( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2.
%F A224268 C = (-1)^m*2^(2*m+1)/Catalan(m) * Product_{n >= 1} ( 1 - (4*m + 3)^2/(4*n + 1)^2 ), for m = 0,1,2,....
%F A224268 C = Integral_{x = 0..1} 1/sqrt(1 + x^4) dx.
%F A224268 C = (1/sqrt(2))*Integral_{x = 0..1} 1/sqrt(1 - x^4) dx.
%F A224268 C = (3/2)*Integral_{x = 0..1} sqrt(1 + x^4) dx - sqrt(2)/2.
%F A224268 C = (1/8)*Integral_{x = 0..1} 1/(x - x^2)^(3/4) dx.
%F A224268 C = Sum_{n >= 0} binomial(-1/2,n)/(4*n + 1) = Sum_{n >= 0} binomial(2*n,n)/4^n * 1/(4*n + 1).
%F A224268 C = (1/2)*Sum_{n >= 0} (-1)^n*binomial(-3/4,n)/(4*n + 1).
%F A224268 Continued fraction: 1 - 1/(5 + 20/(1 + 30/(3 + ... + (4*n)*(4*n + 1)/(1 + (4*n + 1)*(4*n + 2)/(3 + ... ))))).
%F A224268 C = A085565/sqrt(2). C = Pi/(4*A096427). (End)
%F A224268 Equals A093341/2 = A327996^2. - _Hugo Pfoertner_, Oct 31 2024
%e A224268 0.9270373386506859592169251735976300231087994117608834527929640225280...
%t A224268 RealDigits[N[Product[1 - 1/(4 n + 1)^2, {n, 1, Infinity}], 90]][[1]] (* or, by the formula: *) RealDigits[Gamma[1/4]^2/(8 Sqrt[Pi]), 10, 90][[1]]
%o A224268 (PARI) prodnumrat(1 - 1/(4*n+1)^2, 1) \\ _Charles R Greathouse IV_, Feb 07 2025
%Y A224268 Cf. A064853, A085565, A093341, A327996.
%Y A224268 Cf. product(1-1/(4n+r)^2, n>=1): A096427 (r=-1), A112628 (r=0), A179587-1 (r=2).
%K A224268 nonn,cons
%O A224268 0,1
%A A224268 _Bruno Berselli_, Apr 02 2013