OFFSET
0,1
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Bala, Notes on the constants A096427 and A224268.
FORMULA
Equals Gamma(1/4)^2/(8*sqrt(Pi)) = L/(4*sqrt(2)), where L is the Lemniscate constant (A064853).
From Peter Bala, Feb 26 2019: (Start)
C = (Pi/4)*( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2.
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,....
C = Integral_{x = 0..1} 1/sqrt(1 + x^4) dx.
C = (1/sqrt(2))*Integral_{x = 0..1} 1/sqrt(1 - x^4) dx.
C = (3/2)*Integral_{x = 0..1} sqrt(1 + x^4) dx - sqrt(2)/2.
C = (1/8)*Integral_{x = 0..1} 1/(x - x^2)^(3/4) dx.
C = Sum_{n >= 0} binomial(-1/2,n)/(4*n + 1) = Sum_{n >= 0} binomial(2*n,n)/4^n * 1/(4*n + 1).
C = (1/2)*Sum_{n >= 0} (-1)^n*binomial(-3/4,n)/(4*n + 1).
Continued fraction: 1 - 1/(5 + 20/(1 + 30/(3 + ... + (4*n)*(4*n + 1)/(1 + (4*n + 1)*(4*n + 2)/(3 + ... ))))).
EXAMPLE
0.9270373386506859592169251735976300231087994117608834527929640225280...
MATHEMATICA
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]]
CROSSREFS
KEYWORD
AUTHOR
Bruno Berselli, Apr 02 2013
STATUS
approved