cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A175573 Decimal expansion of Pi^(1/4)/Gamma(3/4).

Original entry on oeis.org

1, 0, 8, 6, 4, 3, 4, 8, 1, 1, 2, 1, 3, 3, 0, 8, 0, 1, 4, 5, 7, 5, 3, 1, 6, 1, 2, 1, 5, 1, 0, 2, 2, 3, 4, 5, 7, 0, 7, 0, 2, 0, 5, 7, 0, 7, 2, 4, 5, 2, 1, 8, 8, 8, 5, 9, 2, 0, 7, 9, 0, 3, 1, 5, 9, 8, 1, 8, 5, 6, 7, 3, 2, 2, 6, 7, 1, 0, 9, 7, 9, 5, 9, 6, 0, 5, 6, 1, 6, 1, 8, 4, 8, 9, 6, 7, 9, 7, 6, 4, 0, 3, 7, 4, 1
Offset: 1

Views

Author

R. J. Mathar, Jul 15 2010

Keywords

Comments

Entry 34 a of chapter 11 of Ramanujan's second notebook. Entry 34 b is A085565.

Examples

			1.0864348112133080145753161...

Links

Crossrefs

Cf. A068465, A085565, A092040, A096427, A175574, A247217, A273081, A273082, A273083, A273084, A319332, A327995.

Programs

  • Magma
    C := ComplexField(); [(Pi(C))^(1/4)/Gamma(3/4)]; // G. C. Greubel, Nov 05 2017
  • Maple
    Pi^(1/4)/GAMMA(3/4) ; evalf(%) ;
  • Mathematica
    RealDigits[ Pi^(1/4)/Gamma[3/4], 10, 105][[1]] (* Jean-François Alcover, Jul 04 2013 *)
  • PARI
    Pi^(1/4)/gamma(3/4) \\ G. C. Greubel, Nov 05 2017
  • PARI
    2*suminf(k=0,exp(-Pi)^(k^2))-1 \\ Hugo Pfoertner, Sep 17 2018

Formula

Equals A092040 / A068465.
Equals Sum_{n=-oo..oo} exp(-Pi*n^2), or also EllipticTheta(3, 0, exp(-Pi)). - Jean-François Alcover, Jul 04 2013
Equals sqrt(A175574). - Amiram Eldar, Jul 04 2023
Equals Gamma(1/4)/(sqrt(2)*Pi^(3/4)). - Vaclav Kotesovec, Jul 04 2023
Equals Product_{k>=1} tanh((1/2 + i/2)*Pi*k), i=sqrt(-1). - _Antonio Graciá Llorente, Mar 20 2024
Equals Product_{k>=0} (1/2)*(((k+1/2)/(k+1))^(1/2)+((k+1)/(k+1/2))^(1/2)). - Antonio Graciá Llorente, Jul 23 2024
Equals (1/A096427)^2 (see Spanier and Oldham at p. 258). - Stefano Spezia, Dec 31 2024
Equals 2*A319332 = 1/A327995. - Hugo Pfoertner, Dec 31 2024

A096427 Decimal expansion of 1/(sqrt(2)*G), where G is Gauss's constant A014549.

Original entry on oeis.org

8, 4, 7, 2, 1, 3, 0, 8, 4, 7, 9, 3, 9, 7, 9, 0, 8, 6, 6, 0, 6, 4, 9, 9, 1, 2, 3, 4, 8, 2, 1, 9, 1, 6, 3, 6, 4, 8, 1, 4, 4, 5, 9, 1, 0, 3, 2, 6, 9, 4, 2, 1, 8, 5, 0, 6, 0, 5, 7, 9, 3, 7, 2, 6, 5, 9, 7, 3, 4, 0, 0, 4, 8, 3, 4, 1, 3, 4, 7, 5, 9, 7, 2, 3, 2, 0, 0, 2, 9, 3, 9, 9, 4, 6, 1, 1, 2, 2, 9, 9, 4, 2
Offset: 0

Views

Author

Eric W. Weisstein, Jul 21 2004

Keywords

Comments

Also, decimal expansion of Product_{n>=1} (1-1/(4n-1)^2). - Bruno Berselli, Apr 02 2013

Examples

			0.8472130847939790866064991234821916364814459103269... = agm(1, sqrt(1/2))

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 421.
  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 1, equation 1:7:6 at page 13.

Links

Crossrefs

Cf. A014549, A062539, A224268, A091670 (1/C^2), A175574 (1/C), A293238 (C^2), A053004 (sqrt(2)*C), A327995.

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(3/4)^2/(Sqrt(2)*Sqrt(Pi(R)/2)); // G. C. Greubel, Aug 17 2018
  • Mathematica
    RealDigits[ArithmeticGeometricMean[1, Sqrt[2]]/Sqrt[2], 10, 110][[1]] (* Bruno Berselli, Apr 02 2013 *)
(* From the comment: *) RealDigits[N[Product[1 - 1/(4 n - 1)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)
  • PARI
    { default(realprecision, 20080); x=agm(1, sqrt(1/2)); d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b096427.txt", n, " ", d)); } \\ Harry J. Smith, Apr 15 2009
  • PARI
    agm(1, sqrt(1/2)) \\ Michel Marcus, Jun 09 2019

Formula

Also equals agm(1,1/sqrt(2)) since agm(1,1/b) = (1/b)*agm(1,b). - Gerald McGarvey, Sep 22 2008
From Peter Bala, Feb 26 2019: (Start)
C = Gamma(3/4)^2/sqrt(Pi).
C = 1/( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2.
C = (1/sqrt(2)) * 1/( Sum_{n = -inf..inf} (-1)^n*exp(-Pi*n^2 ) )^2.
Conjecturally, C = (1/sqrt(2)) * 1/( Sum_{n = -inf..inf} exp(-Pi*(n+1/2)^2 ) )^2.
C = ((-1)^m*4^m/binomial(2*m,m)) * Product_{n >= 0} ( 1 - (4*m + 1)^2/(4*n + 3)^2 ), for m = 0,1,2,....
C = 1 - Integral_{x = 0..1} (sqrt(1 + x^4) - 1)/x^2 dx.
C = 1 - Sum_{n >= 1} binomial(1/2,n)/(4*n - 1) = 1 - Sum_{n >= 0} (-1)^n/(4*n + 3)*Catalan(n)/2^(2*n + 1).
Continued fraction: 1 - 1/(3 + 6/(1 + 12/(3 + ... + (4*n - 1)*(4*n - 2)/(1 + 4*n*(4*n - 1)/(3 + ... ))))). (End)
From Peter Bala, Mar 02 2022 : (Start)
C = (2/3)*hypergeom([1/4, 3/4], [7/4], 1)
C = hypergeom([-1/4, 1/4], [3/4], 1).
C = hypergeom([-1/2, -1/4], [3/4], -1). Cf. A053004.
C = (16/21)*hypergeom([-1/4, -3/4], [7/4], 1). (End)
Equals Pi/(sqrt(2)*A062539). - Amiram Eldar, May 04 2022
C = Integral_{x = 0..Pi/2} sqrt(sin(x)*cos(x)) dx. - Adam Hugill, Nov 27 2022
Equals 1/A175574 = sqrt(A293238) = A327995^2. - Hugo Pfoertner, Dec 26 2024

A327996 Decimal expansion of (1/2)*Pi^(3/4)/Gamma(3/4).

Original entry on oeis.org

9, 6, 2, 8, 2, 7, 7, 8, 2, 4, 4, 6, 4, 1, 7, 5, 4, 7, 9, 1, 9, 0, 9, 2, 2, 1, 5, 4, 4, 8, 5, 2, 2, 9, 7, 8, 2, 7, 1, 0, 0, 8, 5, 1, 4, 4, 7, 8, 5, 8, 0, 6, 7, 1, 0, 0, 1, 8, 2, 1, 5, 1, 4, 9, 5, 0, 0, 4, 5, 5, 3, 5, 7, 5, 7, 5, 5, 0, 9, 8, 6, 9, 7, 0, 9, 1
Offset: 0

Views

Author

Peter Luschny, Oct 24 2019

Keywords

Comments

The function df(x) = 2^(x/2)*(2/Pi)^(sin(Pi*x/2)^2/2)*Gamma(x/2+1) interpolates the double factorials A006882 and extends them analytically. df(1/2) is the given constant. Extending also the notation this can be written as (1/2)!! = Pi^(3/4)/(2*(-1/4)!).

Examples

			Equals 0.962827782446417547919092215448522978271008514478580671...

Crossrefs

Cf. A006882, A319332, A327995.

Programs

  • Maple
    Digits := 100: (1/2)*Pi^(3/4)/GAMMA(3/4)*10^86:
ListTools:-Reverse(convert(floor(%), base, 10));
  • Mathematica
    RealDigits[Pi^(3/4)/(2*Gamma[3/4]), 10, 120][[1]] (* Amiram Eldar, May 30 2023 *)
  • PARI
    (1/2)*Pi^(3/4)/gamma(3/4) \\ Michel Marcus, Oct 24 2019

Formula

Equals Pi^(3/4)/(2*(-1/4)!).
From Amiram Eldar, May 30 2023: (Start)
Equals Gamma(1/4)/(2*sqrt(2)*Pi^(1/4)).
Equals A319332 * sqrt(Pi). (End)
Showing 1-3 of 3 results.

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