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 1-6 of 6 results.

A247041 Decimal expansion of zeta(5/2).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Sep 10 2014

Keywords

Comments

Zeta(5/2) appears in the expression of the 6th Madelung constant (A247040).

Examples

			1.3414872572509171797567696933486121366230376295059865...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 77.

Links

Crossrefs

Cf. A059750, A078434, A247040.

Programs

  • Mathematica
    RealDigits[Zeta[5/2], 10, 102] // First
  • PARI
    zeta(5/2) \\ Michel Marcus, Feb 23 2023

A261805 Decimal expansion of M_8, the 8th Madelung constant (negated).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Sep 01 2015

Keywords

Examples

			-2.052466827269271228176337799173383991708377529965582...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 77.

Links

Crossrefs

Cf. A059750, A088537, A085469, A090734, A247040, A261804.

Programs

  • Mathematica
    M8 = (15/(4*Pi^3))*(8*Sqrt[2] - 1)*Zeta[1/2]*Zeta[7/2]; RealDigits[M8, 10, 103] // First
  • PARI
    th4(x)=1+2*sumalt(n=1, (-1)^n*x^n^2)
intnum(x=0, [oo, 1], (th4(exp(-x))^8-1)/sqrt(Pi*x)) \\ Charles R Greathouse IV, Jun 06 2016

Formula

M_8 = (15/(4*Pi^3))*(8*sqrt(2) - 1)*zeta(1/2)*zeta(7/2).

A247042 Decimal expansion of delta_2 (negated), a constant associated with a certain two-dimensional lattice sum.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Sep 10 2014

Keywords

Comments

This constant is named sigma(1/2) in the Borwein reference.

Examples

			-3.900264920001955882845475336604973219209047856477537388...

References

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

Links

Crossrefs

Cf. A088537, A085469, A090734, A247040.

Programs

  • Mathematica
    delta2 = 2*Zeta[1/2]*(Zeta[1/2, 1/4] - Zeta[1/2, 3/4]); RealDigits[delta2, 10, 102] // First
  • PARI
    2*zeta(1/2)*(zetahurwitz(1/2,1/4)-zetahurwitz(1/2,3/4)) \\ Charles R Greathouse IV, Jan 31 2018

Formula

delta_2 = 2*zeta(1/2)*(zeta(1/2, 1/4) - zeta(1/2, 3/4)), where zeta(s,a) gives the generalized Riemann zeta function.

A264156 Decimal expansion of M_5, the 5-dimensional analog of Madelung's constant (negated).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Nov 06 2015

Keywords

Examples

			-1.9093378156187685595201437984336...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 77.

Links

Crossrefs

Cf. A088537, A085469, A090734, A247040, A261805, A264157.

Programs

  • Mathematica
    digits = 32; f[n_, x_] := 1/Sqrt[Pi*x]*(EllipticTheta[4, 0, Exp[-x]]^n - 1); M[5] = NIntegrate[f[5, x], {x, 0, Infinity}, WorkingPrecision -> digits + 5]; RealDigits[M[5], 10, digits] // First

Formula

Equals (1/sqrt(Pi))*Integral_{t=0..oo} ((Sum_{k=-oo..oo} (-1)^k*exp(-k^2*t))^5 - 1)/sqrt(t) dt.

A264157 Decimal expansion of M_7, the 7-dimensional analog of Madelung's constant (negated).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Nov 06 2015

Keywords

Examples

			-2.01240598979798606439503063580430044165678065812192932878490469117330...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 77.

Links

Crossrefs

Cf. A088537, A085469, A090734, A247040, A261805, A264156.

Programs

  • Mathematica
    digits = 32; f[n_, x_] := 1/Sqrt[Pi*x]*(EllipticTheta[4, 0, Exp[-x]]^n - 1); M[7] = NIntegrate[f[7, x], {x, 0, Infinity}, WorkingPrecision -> digits + 5]; RealDigits[M[7], 10, digits] // First
  • PARI
    th4(x)=1+2*sumalt(n=1,(-1)^n*x^n^2)
intnum(x=0,[oo,1], (th4(exp(-x))^7-1)/sqrt(Pi*x)) \\ Charles R Greathouse IV, Jun 06 2016

Formula

Equals (1/sqrt(Pi))*Integral_{t=0..oo} ((Sum_{k=-oo..oo} (-1)^k*exp(-k^2*t))^7-1)/sqrt(t) dt.

Extensions

More terms from Charles R Greathouse IV, Jun 06 2016

A246966 Decimal expansion of H_2, the analog of Madelung's constant for the planar hexagonal lattice.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Sep 10 2014

Keywords

Comments

The ionic hexagonal (triangular) lattice considered here consists of three interpenetrating hexagonal lattices of ions with charges +1, -1, 0. Equivalently, one may consider the honeycomb net consisting of two hexagonal lattices of ions with charges +1 and -1 (the h-BN layer structure). In any case, this lattice sum is based on the nearest neighbor distance (not the length of the period of the ionic crystal structure, which is sqrt(3) times greater). - Andrey Zabolotskiy, Jun 21 2022

Examples

			1.54221972170650525853141576436424529561948...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 78.

Links

Crossrefs

Cf. A088537, A085469, A090734, A247040.

Programs

  • Mathematica
    H2 = (-3 + Sqrt[3])*Zeta[1/2]*((1 - Sqrt[2])*Zeta[1/2, 1/3] + Zeta[1/2, 1/6]); RealDigits[H2, 10, 103] // First
  • PARI
    (sqrt(3)-3)*zeta(1/2)*((1-sqrt(2))*zetahurwitz(1/2, 1/3) + zetahurwitz(1/2, 1/6)) \\ Charles R Greathouse IV, Jan 31 2018

Formula

H_2 = (-3 + sqrt(3))*zeta(1/2)*((1 - sqrt(2))*zeta(1/2, 1/3) + zeta(1/2, 1/6)), where zeta(s,a) gives the generalized Riemann zeta function.
Showing 1-6 of 6 results.

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