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 1-4 of 4 results.

A086231 Decimal expansion of value of Watson's integral.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Jul 12 2003

Keywords

Examples

			1.51638605915197801815601215968142077995538704445226267656698...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 322.

Links

Crossrefs

Cf. A002896, A086230, A242812, A242813, A242814, A242815, A242816, A273086.

Programs

  • Magma
    C := ComplexField(); (Sqrt(3)-1)*(Gamma(1/24)*Gamma(11/24))^2/(32*Pi(C)^3); // G. C. Greubel, Jan 07 2018
  • Maple
    evalf((sqrt(3)-1)*(GAMMA(1/24)*GAMMA(11/24))^2 / (32*Pi^3),120); # Vaclav Kotesovec, Sep 16 2014
  • Mathematica
    RealDigits[ N[ Sqrt[6]/32/Pi^3*Gamma[1/24]*Gamma[5/24]*Gamma[7/24]*Gamma[11/24], 102]][[1]] (* Jean-François Alcover, Nov 12 2012, after Eric W. Weisstein *)
  • PARI
    (sqrt(3)-1)*(gamma(1/24)*gamma(11/24))^2 / (32*Pi^3) \\ Altug Alkan, Apr 13 2016

Formula

Equals (sqrt(3)-1)*(gamma(1/24)*gamma(11/24))^2/(32*Pi^3). - G. C. Greubel, Jan 07 2018
Equals 1/(1 - A086230). - Amiram Eldar, Aug 28 2020
Equals Sum_{k>=0} A002896(k)/36^k. - Vaclav Kotesovec, Apr 23 2023

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).

A273084 Decimal expansion of theta_3(0, exp(-6*Pi)), where theta_3 is the 3rd Jacobi theta function.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, May 14 2016

Keywords

Examples

			1.0000000130248242721598014564243302309067457325414604157511...

Links

Crossrefs

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

Programs

  • Magma
    C := ComplexField(); Sqrt(2+Sqrt(8+6*Sqrt(3)+4*Sqrt(6 +4*Sqrt(3))))*Pi(C)^(1/4)/(2*3^(3/8)*Gamma(3/4)) // G. C. Greubel, Jan 07 2018
  • Maple
    evalf(sqrt(2 + sqrt(8 + 6*sqrt(3) + 4*sqrt(6 + 4*sqrt(3)))) * Pi^(1/4) / (2*3^(3/8) * GAMMA(3/4)), 120);
  • Mathematica
    RealDigits[EllipticTheta[3, 0, Exp[-6*Pi]], 10, 105][[1]]
RealDigits[Sqrt[2 + Sqrt[8 + 6*Sqrt[3] + 4*Sqrt[6 + 4*Sqrt[3]]]] * Pi^(1/4) / (2*3^(3/8) * Gamma[3/4]), 10, 105][[1]]
  • PARI
    th3(x)=1 + 2*suminf(n=1,x^n^2) th3(exp(-6*Pi)) \\ Charles R Greathouse IV, Jun 06 2016

Formula

Equals sqrt(2+sqrt(8+6*sqrt(3)+4*sqrt(6+4*sqrt(3)))) * Pi^(1/4) / (2*3^(3/8)*Gamma(3/4)).
Equals sqrt((A273081^2 + A292888^4/A363018^2)/2). - Vaclav Kotesovec, May 17 2023

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 1-4 of 4 results.

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