A185580 -id:A185580 - 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

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

Original entry on oeis.org

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

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

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A185576 Decimal expansion of Born's basic potential Pi_0.

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

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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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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 ).