A224268 - OEIS

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

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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 + ... ))))).

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

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

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

Cf. A064853, A085565, A093341, A327996.

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

Sequence in context: A248317 A332631 A179638 * A019877 A252898 A336912

Adjacent sequences: A224265 A224266 A224267 * A224269 A224270 A224271

KEYWORD

nonn,cons

AUTHOR

Bruno Berselli, Apr 02 2013

STATUS

approved