A185577 -id:A185577 - cp's OEIS frontend

A185576 Decimal expansion of Born's basic potential Pi_0.

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

Offset: 1

R. J. Mathar, Jan 31 2011

Decimal expansion of Sum'_{m,n,p = -infinity..infinity} 1/(m^2 + n^2 + p^2)^s, analytic continuation to s=1/2. The prime at the sum symbol means the term at m=n=p=0 is omitted.

			2.8372974794806194766659171046...

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

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_0.
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 a(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 a(1).

Cf. A185577, A185578, A185579, A185580, A185581, A185582, A185583.

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]; RealDigits[s[k], 10, digits] // First (* Jean-François Alcover, Sep 10 2014 *)

Equals A085469/3 + A185577 + A185578.

More terms from Jean-François Alcover, Sep 10 2014

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

A181152 Decimal expansion of Madelung constant (negated) for the CsCl structure.

Original entry on oeis.org

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

Offset: 1

Leslie Glasser, Jan 24 2011

This is often quoted for a different lattice constant and multiplied by 2/sqrt(3) = 1.1547... = 10*A020832, which gives 1.76267...*1.1547... = 2.03536151... given in Zucker's Table 5 as the alpha for the CsCl structure, and by Sakamoto as the M_d for the B2 lattice. Given Zucker's b(1) = 0.774386141424002815... = A185577, this constant here is sqrt(3)*(3*b(1)+A085469)/4. - R. J. Mathar, Jan 28 2011

The CsCl structure consists of two interpenetrating simple cubic lattices of ions with charges +1 and -1, together occupying all the sites of the body-centered cubic lattice. - Andrey Zabolotskiy, Oct 21 2019

Leslie Glasser, Solid-State Energetics and Electrostatics: Madelung Constants and Madelung Energies, Inorg. Chem., 2012, 51 (4), 2420-2424.
Y. Sakamoto, Madelung Constants of Simple Crystals Expressed in Terms of Born's Basic Potentials of 15 Figures, Journal of Chemical Physics, 28 (1958), 164-165. Errata: J. Chem. Phys, 28 (1958), 733; J. Chem. Phys, 28 (1958), 1253.
Nicolas Tavernier, Gian Luigi Bendazzoli, Véronique Brumas, Stefano Evangelisti, and J. A. Berger, Clifford boundary conditions: a simple direct-sum evaluation of Madelung constants, J. Phys. Chem. Lett., 11 (2020), 7090-7095; arXiv:2006.01259 [physics.comp-ph], 2020.
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.
Wikipedia, Madelung constant

Cf. A085469, A088537, A090734.

digits = 105;
m0 = 50; (* initial number of terms *)
dm = 10; (* number of terms increment *)
dd = 10; (* precision excess *)
Clear[f];
f[n_, p_] := f[n, p] = (s = Sqrt[n^2 + p^2]; ((2 + (-1)^n) Csch[s*Pi])/s // N[#, digits + dd]&);
f[m_] := f[m] = Pi/2 - (7 Log[2])/2 + 4 Sum[f[n, p], {n, 1, m}, {p, 1, m}];
f[m = m0];
f[m += dm];
While[Abs[f[m] - f[m - dm]] > 10^(-digits - dd), Print["f(", m, ") = ", f[m]]; m += dm];
A185577 = f[m];
Clear[g];
g[m_] := g[m] = 12 Pi Sum[Sech[(Pi/2) Sqrt[(2 j + 1)^2 + (2 k + 1)^2]]^2, {j, 0, m}, {k, 0, m}] // N[#, digits + dd]&;
g[m = m0];
g[m += dm];
While[Abs[g[m] - g[m - dm]] > 10^(-digits - dd), Print["g(", m, ") = ", g[m]]; m += dm];
A085469 = g[m];
A181152 = Sqrt[3] (A085469 - 3 A185577)/4;
RealDigits[A181152, 10, digits][[1]] (* Jean-François Alcover, May 07 2021 *)

More terms (using the above comment from R. J. Mathar and terms from the b-files for A085469 and A185577) from Jon E. Schoenfield, Mar 10 2018
Definition corrected by Andrey Zabolotskiy, Oct 21 2019
a(88)-a(105) from Jean-François Alcover, May 07 2021

A182565 Decimal expansion of Madelung constant (negated) for cuprous oxide Cu_2O.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, May 05 2012

			-4.4424752098389...

Leslie Glasser, Solid-State Energetics and Electrostatics: Madelung Constants and Madelung Energies, Inorg. Chem., 2012, 51 (4), 2420-2424.
Yosio Sakamoto, Madelung Constants of Simple Crystals Expressed in Terms of Born's Basic Potentials of 15 Figure, The Journal of Chemical Physics 28, 164 (1958). See Table III, row C3, column M_r.
Wikipedia, Copper(I) oxide.

Cf. A085469, A181152, A182566, A182567, A185577, A185578.

Equals (sqrt(3)/4) * (2 * A085469 + 3 * (A185577 + A185578)). [Sakamoto] - Andrey Zabolotskiy, Oct 12 2023

More terms using the formula added by Andrey Zabolotskiy, Oct 12 2023

cp's OEIS Frontend

A185576 Decimal expansion of Born's basic potential Pi_0.

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

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

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

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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 ).

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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).

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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).

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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 ).