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 1-7 of 7 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

A247217 Decimal expansion of theta_3(0, exp(-2*Pi)), where theta_3 is the 3rd Jacobi theta function.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Nov 26 2014

Keywords

Examples

			1.0037348854877390910476795950669538662079943324445194...

Links

Crossrefs

Cf. A000122, A175573 (theta_3(0, exp(-Pi))), A273081, A273082, A273083, A273084.
Cf. A273087, A273086.

Programs

  • Magma
    C := ComplexField(); (4*Pi(C)*Sqrt(2) + 6*Pi(C))^(1/4)/(2*Gamma(3/4)); // G. C. Greubel, Jan 07 2018
  • Mathematica
    RealDigits[(4*Pi*Sqrt[2] + 6*Pi)^(1/4)/(2*Gamma[3/4]), 10, 102] // First
  • PARI
    (4*Pi*sqrt(2) + 6*Pi)^(1/4)/(2*gamma(3/4)) \\ Michel Marcus, Nov 26 2014

Formula

Equals (4*Pi*sqrt(2) + 6*Pi)^(1/4)/(2*Gamma(3/4)).
Equals Sum_{n=-infinity..infinity} exp(-2*Pi*n^2).

A273081 Decimal expansion of theta_3(0, exp(-3*Pi)), where theta_3 is the 3rd Jacobi theta function.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, May 14 2016

Keywords

Examples

			1.0001613990351406940215020703893995738875083912423752893728...

Links

Crossrefs

Cf. A175573, A247217, A273082, A273083, A273084.

Programs

  • Magma
    C := ComplexField(); (1+2/Sqrt(3))^(1/4)*Pi(C)^(1/4)/(3^(1/4) *Gamma(3/4)) // G. C. Greubel, Jan 07 2018
  • Maple
    evalf((1 + 2/sqrt(3))^(1/4) * Pi^(1/4) / (3^(1/4) * GAMMA(3/4)), 120);
  • Mathematica
    RealDigits[EllipticTheta[3, 0, Exp[-3*Pi]], 10, 105][[1]]
RealDigits[(1 + 2/Sqrt[3])^(1/4) * Pi^(1/4) / (3^(1/4) * Gamma[3/4]), 10, 105][[1]]
  • PARI
    sqrtn((2/sqrt(3)+1)*Pi/3,4)/gamma(3/4) \\ Charles R Greathouse IV, Jun 06 2016

Formula

Equals (1 + 2/sqrt(3))^(1/4) * Pi^(1/4) / (3^(1/4) * Gamma(3/4)).

A273082 Decimal expansion of theta_3(0, exp(-4*Pi)), where theta_3 is the 3rd Jacobi theta function.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, May 14 2016

Keywords

Examples

			1.00000697468471241799127935745572277338608481181934395967...

Links

Crossrefs

Cf. A175573, A247217, A273081, A273083, A273084.

Programs

  • Magma
    C := ComplexField(); ((2+2^(3/4))/4)*Pi(C)^(1/4)/Gamma(3/4) // G. C. Greubel, Jan 07 2018
  • Maple
    evalf((2+2^(3/4))/4*Pi^(1/4)/GAMMA(3/4), 120);
evalf((1 + 2^(1/4)) * GAMMA(1/4) / (2^(7/4)*Pi^(3/4)), 120);
  • Mathematica
    RealDigits[EllipticTheta[3, 0, Exp[-4*Pi]], 10, 105][[1]]
RealDigits[(2 + 2^(3/4))/4 * Pi^(1/4) / Gamma[3/4], 10, 105][[1]]
RealDigits[(1 + 2^(1/4)) * Gamma[1/4] / (2^(7/4)*Pi^(3/4)), 10, 105][[1]]
  • PARI
    (2^.25+1)*gamma(1/4)/sqrtn(128*Pi^3,4) \\ Charles R Greathouse IV, Jun 06 2016

Formula

Equals (2 + 2^(3/4))/4 * Pi^(1/4) / Gamma(3/4).
Equals (1 + 2^(1/4)) * Gamma(1/4) / (2^(7/4)*Pi^(3/4)).
Equals sqrt((A247217^2 + A259149^4/A259150^2)/2). - Vaclav Kotesovec, May 17 2023

A273083 Decimal expansion of theta_3(0, exp(-5*Pi)), where theta_3 is the 3rd Jacobi theta function.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, May 14 2016

Keywords

Examples

			1.0000003014034550780129221506549039080802236178954948667347...

Links

Crossrefs

Cf. A175573, A247217, A273081, A273082, A273084.

Programs

  • Magma
    C := ComplexField(); Sqrt(2/5 + 1/Sqrt(5))*Pi(C)^(1/4)/Gamma(3/4) // G. C. Greubel, Jan 07 2018
  • Maple
    evalf(sqrt(2/5 + 1/sqrt(5)) * Pi^(1/4)/GAMMA(3/4), 120);
  • Mathematica
    RealDigits[EllipticTheta[3, 0, Exp[-5*Pi]], 10, 105][[1]]
RealDigits[Sqrt[2/5 + 1/Sqrt[5]] * Pi^(1/4)/Gamma[3/4], 10, 105][[1]]
  • PARI
    th3(x)=1 + 2*suminf(n=1,x^n^2)
th3(exp(-5*Pi)) \\ Charles R Greathouse IV, Jun 06 2016

Formula

Equals sqrt(2/5 + 1/sqrt(5)) * Pi^(1/4)/Gamma(3/4).

A273086 Decimal expansion of theta_3(0, exp(-sqrt(6)*Pi)), where theta_3 is the 3rd Jacobi theta function.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, May 14 2016

Keywords

Examples

			1.0009099218872567629192860041215666718045881467303013308592...

Links

Crossrefs

Cf. A175573, A247217, A273081, A273082, A273083, A273084, A086231, A273087.

Programs

  • Maple
    evalf(((6 + sqrt(6*(3 + 2*sqrt(2)))) * GAMMA(1/24) * GAMMA(5/24) * GAMMA(7/24) * GAMMA(11/24))^(1/4) / (2*6^(3/8)*Pi^(3/4)), 120);
evalf((4 - sqrt(2) + sqrt(6))^(1/4) * sqrt(GAMMA(1/24)*GAMMA(11/24)) / (2^(3/2)*3^(3/8)*Pi^(3/4)), 120);
  • Mathematica
    RealDigits[EllipticTheta[3, 0, Exp[-Sqrt[6]*Pi]], 10, 105][[1]]
RealDigits[((6 + Sqrt[6*(3 + 2*Sqrt[2])]) * Gamma[1/24] * Gamma[5/24] * Gamma[7/24] * Gamma[11/24])^(1/4) / (2*6^(3/8)*Pi^(3/4)), 10, 105][[1]]
RealDigits[(4 - Sqrt[2] + Sqrt[6])^(1/4) * Sqrt[Gamma[1/24]*Gamma[11/24]] / (2^(3/2)*3^(3/8)*Pi^(3/4)), 10, 105][[1]]
  • PARI
    th3(x)=1 + 2*suminf(n=1,x^n^2)
th3(exp(-sqrt(6)*Pi)) \\ Charles R Greathouse IV, Jun 06 2016

Formula

Equals ((6 + sqrt(6*(3 + 2*sqrt(2)))) * Gamma(1/24) * Gamma(5/24) * Gamma(7/24) * Gamma(11/24))^(1/4) / (2*6^(3/8)*Pi^(3/4)).
Equals (4 - sqrt(2) + sqrt(6))^(1/4) * sqrt(Gamma(1/24)*Gamma(11/24)) / (2^(3/2)*3^(3/8)*Pi^(3/4)).

A273087 Decimal expansion of theta_3(0, exp(-sqrt(2)*Pi)), where theta_3 is the 3rd Jacobi theta function.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, May 14 2016

Keywords

Examples

			1.0235239993410058634977986567249718525649146079487847418721...

Links

Crossrefs

Cf. A175573, A247217, A273081, A273082, A273083, A273084, A273086.

Programs

  • Magma
    C := ComplexField(); Gamma(1/8)/(2^(9/8)*Sqrt(Pi(C)*Gamma(1/4))) // G. C. Greubel, Jan 07 2018
  • Maple
    evalf(GAMMA(1/8)/(2^(9/8)*sqrt(Pi*GAMMA(1/4))), 120);
  • Mathematica
    RealDigits[EllipticTheta[3, 0, Exp[-Sqrt[2]*Pi]], 10, 105][[1]]
RealDigits[Gamma[1/8]/(2^(9/8)*Sqrt[Pi*Gamma[1/4]]), 10, 105][[1]]
  • PARI
    th3(x)=1 + 2*suminf(n=1,x^n^2)
th3(exp(-sqrt(2)*Pi)) \\ Charles R Greathouse IV, Jun 06 2016

Formula

Equals Gamma(1/8)/(2^(9/8)*sqrt(Pi*Gamma(1/4))).
Showing 1-7 of 7 results.

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