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.

A247043 Decimal expansion of gamma_2, a lattice sum constant, analog of Euler's constant for two-dimensional lattices.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Sep 10 2014

Keywords

Examples

			-0.670908307882478806085271599253853426816260971797672535...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 80.

Links

Crossrefs

Cf. A247042.

Programs

  • Mathematica
    delta2 = 2*Zeta[1/2]*(Zeta[1/2, 1/4] - Zeta[1/2, 3/4]); gamma2 = (1/4)*(delta2 + 2*Log[(Sqrt[2] + 1)/(Sqrt[2] - 1)] - 4*EulerGamma); RealDigits[gamma2, 10, 103] // First
  • PARI
    (2*zeta(1/2)*(zetahurwitz(1/2, 1/4)-zetahurwitz(1/2, 3/4)) + 2*log((sqrt(2) + 1)/(sqrt(2) - 1)))/4 - Euler \\ Charles R Greathouse IV, Jan 31 2018

Formula

gamma_2 = (1/4)*(delta_2 + 2*log((sqrt(2) + 1)/(sqrt(2) - 1)) - 4*EulerGamma), where delta_2 is A247042.

A247046 Decimal expansion of delta_3, a constant associated with a certain 3-dimensional lattice sum.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Sep 10 2014

Keywords

Examples

			-2.313698703882320603588809874061155008356357135595965962...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 79.

Links

Crossrefs

Cf. A088537, A185576, A247042.

Programs

  • Mathematica
    digits = 100; k0 = 10; dk = 10; Clear[s]; s[k_] := s[k] = 7*(Pi/6) - 19/2*Log[2] + 4*Sum[(3 + 3*(-1)^m + (-1)^(m + n))*Csch[Pi*Sqrt[m^2 + n^2]]/Sqrt[m^2 + n^2], {m, 1, k}, {n, 1, k}] // N[#, digits + 10] &; s[k0]; s[k = k0 + dk]; While[RealDigits[s[k], 10, digits + 5][[1]] != RealDigits[s[k - dk], 10, digits + 5][[1]], Print["s(", k, ") = ", s[k]]; k = k + dk]; Pi0 = s[k]; delta3 = Pi0 + Pi/6; RealDigits[delta3, 10, digits] // First

Formula

Pi_0 + Pi/6, where Pi_0 is A185576.

A247277 Decimal expansion of gamma_3, a lattice sum constant, analog of Euler's constant for 3-dimensional lattices.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Sep 11 2014

Keywords

Examples

			0.58174804565972267655489926584685317714602246563144492431364...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 80.

Links

Crossrefs

Cf. A247042 (delta_2), A247043 (gamma_2), A247046 (delta_3).

Programs

  • Mathematica
    digits = 100; k0 = 10; dk = 10; Clear[s]; s[k_] := s[k] = 7*(Pi/6) - 19/2*Log[2] + 4*Sum[(3 + 3*(-1)^m + (-1)^(m + n))*Csch[Pi*Sqrt[m^2 + n^2]]/Sqrt[m^2 + n^2], {m, 1, k}, {n, 1, k}] // N[#, digits + 10] &; s[k0]; s[k = k0 + dk]; While[RealDigits[s[k], 10, digits + 5][[1]] != RealDigits[s[k - dk], 10, digits + 5][[1]], k = k + dk]; Pi0 = s[k]; delta2 = 2*Zeta[1/2]*(Zeta[1/2, 1/4] - Zeta[1/2, 3/4]); delta3 = Pi0 + Pi/6; gamma2 = (1/4)*(delta2 + 2*Log[(Sqrt[2] + 1)/(Sqrt[2] - 1)] - 4*EulerGamma); gamma3 = (1/8)*(delta3 + 3*(- Pi/6 + Log[(Sqrt[3] + 1)/(Sqrt[3] - 1)]) - 12*gamma2 - 6*EulerGamma); RealDigits[gamma3, 10, 102] // First

Formula

gamma_3 = (1/8)*(delta_3 + 3*(- Pi/6 + log((sqrt(3) + 1)/(sqrt(3) - 1))) - 12*gamma_2 - 6*EulerGamma).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.