A185576 -id:A185576 - cp's OEIS frontend

A185577 Decimal expansion of Sum'_{m,n,p = -infinity..infinity} (-1)^m/sqrt(m^2 + n^2 + p^2), negated.

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

Offset: 0

R. J. Mathar, Jan 31 2011

The prime at the sum symbol means that the term at m=n=p=0 is omitted.

			0.77438614142400281521275138640678...

Y. Sakamoto, Madelung constants of simple crystals expressed in terms of Born's basic potentials of 15 figures, J. Chem. Phys. 28 (1958) 164, variable Pi_2.
I. J. Zucker, Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures, J. Phys. A: Math. Gen. 8 (11) (1975) 1734, variable b(1).
I. J. Zucker, Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants, J. Phys. A: Math. Gen. 9 (4) (1976) 499, variable b(1).

Cf. A185576, A185578, A185579, A185580, A185581, A185582, A185583.

digits = 105; Clear[f]; f[n_, p_] := f[n, p] =(s = Sqrt[n^2 + p^2]; ((2 + (-1)^n)*Csch[s*Pi])/s // N[#, digits+10]&); f[m_] := f[m] = Pi/2 - (7*Log[2])/2 + 4*Sum[f[n, p], {n, 1, m}, {p, 1, m}] // RealDigits[#, 10, digits+10]& // First; f[0]; f[m=10]; While[f[m] != f[m-10], Print[m]; m = m+10]; f[m][[1 ;; digits]] (* Jean-François Alcover, Feb 20 2013 *)

sqrt(3)*(3*(this value) + A085469)/4 = A181152.

Equals Pi/2 - 7*log(2)/2 + 4*Sum_{n>=1, p>=1} (2+(-1)^n) *cosech(d*Pi)/d with d = sqrt(n^2 + p^2).

More terms from Jean-François Alcover, Feb 20 2013

A185578 Decimal expansion of Sum'_{m,n,p = -infinity .. infinity} (-1)^(m + n)/sqrt(m^2 + n^2 + p^2), negated.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 31 2011

The prime at the sum symbol means the term at m=n=p=0 is omitted.

			1.48038980651222259790776170...

Y. Sakamoto, Madelung constants of simple crystals expressed in terms of Born's basic potentials of 15 figures, J. Chem. Phys. 28 (1958) 164, variable Pi_3.
I. J. Zucker, Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures, J. Phys. A: Math. Gen. 8 (11) (1975) 1734, variable c(1).
I. J. Zucker, Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants, J. Phys. A: Math. Gen. 9 (4) (1976) 499, variable c(1).

Cf. A185576, A185577, A185579, A185580, A185581, A185582, A185583.

digits = 105; Clear[f]; f[n_, p_] := f[n, p] = (s = Sqrt[n^2 + p^2]; ((1 + (-1)^n + (-1)^(n + p))*Csch[s*Pi])/s // N[#, digits+10]&); f[m_] := f[m] = Pi/2 - 9*Log[2]/2 + 4*Sum[f[n, p], {n, 1, m}, {p, 1, m}] // RealDigits[#, 10, digits + 10]& // First; f[0]; f[m=10]; While[ f[m] != f[m-10], Print[m]; m = m+10]; f[m][[1 ;; digits]] (* Jean-François Alcover, Feb 20 2013 *)

Equals Pi/2 - 9*log(2)/2 + 4*Sum_{p>=1, n>=1} (1+(-1)^n+(-1)^(n+p))*cosech(d*Pi)/d where d = sqrt(n^2 + p^2).

More terms from Jean-François Alcover, Feb 20 2013

A185579 Decimal expansion of Sum_{m,n,p = -infinity..infinity} (-1)^m/sqrt(m^2 + (n-1/2)^2 + (p-1/2)^2).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 31 2011

			1.5401709018555436174364666638648...

I. J. Zucker, Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures, J. Phys. A: Math. Gen. 8 (11) (1975) 1734, variable e(1).
I. J. Zucker, Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants, J. Phys. A: Math. Gen. 9 (4) (1976) 499, variable e(1).

Cf. A185576, A185577, A185578, A185580, A185581, A185582, A185583.

digits = 105; Clear[f]; f[n_, p_] := f[n, p] = (d = Sqrt[n^2 + (p - 1/2)^2]; (-1)^n*(Csch[d*Pi]/d) // N[#, digits+10]&); f[m_] := f[m] = 2*Log[1 + Sqrt[2]] + 4*Sum[f[n, p], {n, 1, m}, {p, 1, m}] // RealDigits[#, 10, digits+10]& // First; f[0]; f[m = 10]; While[f[m] != f[m-10], Print[m]; m = m+10]; f[m][[1 ;; digits]] (* Jean-François Alcover, Feb 21 2013 *)

Equals 2*log(1+sqrt(2)) + 4*Sum_{n>=1, p>=1} (-1)^n*cosech(d*Pi)/d where d = sqrt(n^2 + (p-1/2)^2).

More terms from Jean-François Alcover, Feb 21 2013

A185580 Decimal expansion of Sum_{m,n,p = -infinity..infinity} (-1)^(m+n)/sqrt( m^2 + n^2 + (p-1/2)^2 ).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 31 2011

I. J. Zucker, Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures, J. Phys. A: Math. Gen. 8 (11) (1975) 1734, variable f(1).
I. J. Zucker, Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants, J. Phys. A: Math. Gen. 9 (4) (1976) 499, variable f(1).

Cf. A185576, A185577, A185578, A185579, A185581, A185582, A185583.

digits = 105; Clear[f]; f[n_, p_] := f[n, p] = (d = Sqrt[(n - 1/2)^2 + (p - 1/2)^2]; (Csch[d*Pi]/d) // N[#, digits + 10] &); f[m_] := f[m] = 4*Sum[f[n, p], {n, 1, m}, {p, 1, m}] // RealDigits[#, 10, digits + 10] & // First; f[0]; f[m = 10]; While[f[m] != f[m - 10], Print[m]; m = m + 10]; f[m][[1 ;; digits]] (* Jean-François Alcover, Feb 21 2013 *)

Equals 4*Sum_{n>=1, p>=1} cosech(d*Pi)/d where d = sqrt((n-1/2)^2 + (p-1/2)^2).

More terms from Jean-François Alcover, Feb 21 2013

A185581 Decimal expansion of 8Sum_{m,n,p = -infinity..infinity} (-1)^(m + n + p)/ sqrt( (2m-1/2)^2+(2n-1/2)^2+(2p-1/2)^2 ).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 31 2011

The formula for g(1) in the 1976 paper on page 503 is a factor 2 too large.

			2.533557404433121025294862795718...

I. J. Zucker, Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures, J. Phys. A: Math. Gen. 8 (11) (1975) 1734, variable g(1).
I. J. Zucker, Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants, J. Phys. A: Math. Gen. 9 (4) (1976) 499, variable g(1).

Cf. A185576, A185577, A185578, A185579, A185580, A185582, A185583.

digits = 105; Clear[f]; f[n_, p_] := f[n, p] = (d = Sqrt[(2 n - 1/2)^2/2 + (p - 1/2)^2]; (-1)^n*(Csch[d*Pi]/d) // N[#, digits + 10] &); f[m_] := f[m] = 2 Sqrt[2]*Sum[f[n, p], {n, -m, m}, {p, -m, m}] // RealDigits[#, 10, digits + 10] & // First; f[0]; f[m = 10]; While[ f[m] != f[m - 10], Print[m]; m = m + 10]; f[m][[1 ;; digits]] (* Jean-François Alcover, Feb 21 2013 *)

Equals 2*sqrt(2)*Sum_{n,p = -infinity..infinity} (-1)^n*cosech(d*Pi)/d where d = sqrt( (2*n-1/2)^2/2 + (p-1/2)^2 ).

More terms from Jean-François Alcover, Feb 21 2013

A185582 Decimal expansion of Sum_{m,n,p = -infinity..infinity} 4*(-1)^(m+p)/sqrt(m^2 + (2n-1/2)^2 + (2p-1/2)^2).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 31 2011

The defining equation (3.10i) on page 1738 of the 1975 paper has a typo (m for n).

The value in Table 4 (page 1742, column h(2s)) seems to have two digits swapped.

			3.634899011049148711368042992007815327990...

I. J. Zucker, Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures, J. Phys. A: Math. Gen. 8 (11) (1975) 1734, variable h(1).
I. J. Zucker, Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants, J. Phys. A: Math. Gen. 9 (4) (1976) 499, variable h(1).

Cf. A185576, A185577, A185578, A185579, A185580, A185581, A185583.

digits = 105; Clear[f]; f[n_, p_] := f[n, p] = (d = Sqrt[2 n^2 + (p - 1/2)^2]; (Csch[d*Pi]/d) // N[#, digits + 10] &); f[m_] := f[m] = 4 Log[1 + Sqrt[2]] + 8*Sum[f[n, p], {n, 1, m}, {p, 1, m}] // RealDigits[#, 10, digits + 10] & // First; f[0]; f[m = 10]; While[ f[m] != f[m - 10], Print[m]; m = m + 10]; f[m][[1 ;; digits]] (* Jean-François Alcover, Feb 21 2013 *)

Equals 4*log(1+sqrt(2)) + 8*Sum_{n>=1, p>=1} cosech(d*Pi)/d where d = sqrt(2*n^2 + (p-1/2)^2).

More terms from Jean-François Alcover, Feb 21 2013

A185583 Decimal expansion of Sum_{m,n,p = -infinity..infinity} 4*(-1)^(m + n + p)/sqrt(m^2 + (2n-1/2)^2 + (2p-1/2)^2).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 31 2011

			1.285846549754779458631385161116...

I. J. Zucker, Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants, J. Phys. A: Math. Gen. 9 (4) (1976) 499, variable j(1).
I. J. Zucker, Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures, J. Phys. A: Math. Gen. 8 (11) (1975) 1734, variable j(1).

Cf. A185576, A185577, A185578, A185579, A185580, A185581, A185582.

digits = 105; Clear[f]; f[n_, p_] := f[n, p] = (d = Sqrt[(n - 1/2)^2 + 2*(p - 1/2)^2]; (Csch[d*Pi]/d) // N[#, digits + 10] &); f[m_] := f[m] = 8*Sum[f[n, p], {n, 1, m}, {p, 1, m}] // RealDigits[#, 10, digits + 10] & // First; f[0]; f[m = 10]; While[ f[m] != f[m - 10], Print[m]; m = m + 10]; f[m][[1 ;; digits]] (* Jean-François Alcover, Feb 21 2013 *)

Equals 8*Sum_{n>=1, p>=1} cosech(d*Pi)/d where d = sqrt((n-1/2)^2 + 2*(p-1/2)^2).

More terms from Jean-François Alcover, Feb 21 2013

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

Jean-François Alcover, Sep 10 2014

			-2.313698703882320603588809874061155008356357135595965962...

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

Eric Weisstein's World of Mathematics, Lattice Sum.

Cf. A088537, A185576, A247042.

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

Pi_0 + Pi/6, where Pi_0 is A185576.

A381701 Decimal expansion of the universal aspect ratio, also called the magic box length ratio, L_z/L_x = L_z/L_y, for which the finite-size error of the self-diffusion coefficient vanishes.

Original entry on oeis.org

2, 7, 9, 3, 3, 5, 9, 6, 4, 9

Offset: 1

Alex Eduardo Delhumeau, Mar 04 2025

Derived by Kikugawa et al. (2015) for a rod-shaped rectangular box (box with lengths L_x = L_y <= L_z) with periodic boundary conditions. The self-diffusion coefficient in the x (and y) direction of a monoatomic Lennard-Jones fluid, calculated from molecular dynamics simulation using the Einstein-Helfand formula, D_xx ( = D_yy), becomes system-size independent and represents the true self-diffusion coefficient, D_0.

Based on the expression for the finite-size correction to the self-diffusion coefficient derived from hydrodynamic theory by B. Dünweg and K. Kremer (1993) and greatly popularized by I.-C. Yeh and G. Hummer (2004). Computed to nine decimal places by J. Busch and D. Paschek (2023).

			2.793359649...

J. Busch and D. Paschek, OrthoBoXY: A Simple Way to Compute True Self-Diffusion Coefficients from MD Simulations with Periodic Boundary Conditions without Prior Knowledge of the Viscosity. J. Phys. Chem. B 127 (2023), 7983-7987.
B. Dünweg and K. Kremer, Molecular dynamics simulation of a polymer chain in solution. J. Chem. Phys. 99 (1993), 6983-6997.
G. Kikugawa, T. Nakano, and T. Ohara, Hydrodynamic consideration of the finite size effect on the self-diffusion coefficient in a periodic rectangular parallelipiped system. J. Chem. Phys. 143 (2015), 024507.
I.-C. Yeh and G. Hummer, System-Size Dependence of Diffusion Coefficients and Viscosities from Molecular Dynamics Simulations with Periodic Boundary Conditions. J. Phys. Chem. B 108 (2004), 15873-15879.

Cf. A185576.

cp's OEIS Frontend

A185577 Decimal expansion of Sum'_{m,n,p = -infinity..infinity} (-1)^m/sqrt(m^2 + n^2 + p^2), negated.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A185578 Decimal expansion of Sum'_{m,n,p = -infinity .. infinity} (-1)^(m + n)/sqrt(m^2 + n^2 + p^2), negated.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A185579 Decimal expansion of Sum_{m,n,p = -infinity..infinity} (-1)^m/sqrt(m^2 + (n-1/2)^2 + (p-1/2)^2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A185580 Decimal expansion of Sum_{m,n,p = -infinity..infinity} (-1)^(m+n)/sqrt( m^2 + n^2 + (p-1/2)^2 ).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A185581 Decimal expansion of 8*Sum_{m,n,p = -infinity..infinity} (-1)^(m + n + p)/ sqrt( (2*m-1/2)^2+(2*n-1/2)^2+(2*p-1/2)^2 ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A185582 Decimal expansion of Sum_{m,n,p = -infinity..infinity} 4*(-1)^(m+p)/sqrt(m^2 + (2n-1/2)^2 + (2p-1/2)^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A185583 Decimal expansion of Sum_{m,n,p = -infinity..infinity} 4*(-1)^(m + n + p)/sqrt(m^2 + (2n-1/2)^2 + (2p-1/2)^2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A185581 Decimal expansion of 8Sum_{m,n,p = -infinity..infinity} (-1)^(m + n + p)/ sqrt( (2m-1/2)^2+(2n-1/2)^2+(2p-1/2)^2 ).