A224268 Decimal expansion of Product_{n>=1} (1 - 1/(4n+1)^2).

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

Offset: 0

Author

Bruno Berselli, Apr 02 2013

Keywords

• nonn
• cons

Examples

			0.9270373386506859592169251735976300231087994117608834527929640225280...
		

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.

Crossrefs

Cf. A064853, A085565, A093341, A327996.

Cf. product(1-1/(4n+r)^2, n>=1): A096427 (r=-1), A112628 (r=0), A179587-1 (r=2).

Programs

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]]

PARI

prodnumrat(1 - 1/(4*n+1)^2, 1) \\ Charles R Greathouse IV, Feb 07 2025

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 + ... ))))).

C = A085565/sqrt(2). C = Pi/(4*A096427). (End)

Equals A093341/2 = A327996^2. - Hugo Pfoertner, Oct 31 2024