A085469 -id:A085469 - cp's OEIS frontend

A005875 Theta series of simple cubic lattice; also number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed).

1, 6, 12, 8, 6, 24, 24, 0, 12, 30, 24, 24, 8, 24, 48, 0, 6, 48, 36, 24, 24, 48, 24, 0, 24, 30, 72, 32, 0, 72, 48, 0, 12, 48, 48, 48, 30, 24, 72, 0, 24, 96, 48, 24, 24, 72, 48, 0, 8, 54, 84, 48, 24, 72, 96, 0, 48, 48, 24, 72, 0, 72, 96, 0, 6, 96, 96, 24, 48, 96, 48, 0, 36, 48, 120

Offset: 0

N. J. A. Sloane

Number of ordered triples (i, j, k) of integers such that n = i^2 + j^2 + k^2.
The Madelung Coulomb energy for alternating unit charges in the simple cubic lattice is Sum_{n>=1} (-1)^n*a(n)/sqrt(n) = -A085469. - R. J. Mathar, Apr 29 2006
a(A004215(k))=0 for k=1,2,3,... but no other elements of {a(n)} are zero. - Graeme McRae, Jan 15 2007

			Order and signs are taken into account: a(1) = 6 from 1 = (+-1)^2 + 0^2 + 0^2, a(2) = 12 from 2 = (+-1)^2 + (+-1)^2 + 0^2; a(3) = 8 from 3 = (+-1)^2 + (+-1)^2 + (+-1)^2, etc.
G.f. =  1 + 6*q + 12*q^2 + 8*q^3 + 6*q^4 + 24*q^5 + 24*q^6 + 12*q^8 + 30*q^9 + 24*q^10 + ...

H. Cohen, Number Theory, Vol. 1: Tools and Diophantine Equations, Springer-Verlag, 2007, p. 317.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.
H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th ed., 1999, Chapter V.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 109.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 54.
L. Kronecker, Crelle, Vol. LVII (1860), p. 248; Werke, Vol. IV, p. 188.
C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman and Hall, 2006, p. 43.
T. Nagell, Introduction to Number Theory, Wiley, 1951, p. 194.
W. Sierpiński, 1925. Teorja Liczb. pp. 1-410 (p.61).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 338, Eq. (B').

T. D. Noe, Table of n, a(n) for n = 0..10000
George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Lemma 2.1.
P. T. Bateman, On the representations of a number as the sum of three squares, Trans. Amer. Math. Soc. 71 (1951), 70-101.
S. Bhargava and C. Adiga, A basic bilateral series summation formula and its applications, Integral Transforms and Special Functions, 2 (1994), 165-184.
J. M. Borwein, K-K S. Choi, On Dirichlet series for sums of squares, Raman. J. 7 (2003) 95-127
S. K. K. Choi, A. V. Kumchev, and R. Osburn, On sums of three squares, arXiv:math/0502007 [math.NT], 2005.
M. Doring, J. Haidenbauer, U.-G. Meissner, and A. Rusetsky, Dynamical coupled-channel approaches on a momentum lattice, arXiv:1108.0676 [hep-lat], 2011.
J. A. Ewell, Recursive determination of the enumerator for sums of three squares, Internat. J. Math. and Math. Sci, 24 (2000), 529-532.
O. Fraser and B. Gordon, On representing a square as the sum of three squares, Amer. Math. Monthly, 76 (1969), 922-923.
M. D. Hirschhorn and J. A. Sellers, On Representations of a Number as a Sum of Three Squares, Discrete Mathematics 199 (1999), 85-101.
M. D. Hirschhorn and J. A. Sellers, On Representations Of A Number As A Sum Of Three Squares
A. Martinez Torres, L. R. Dai, C. Koren, D. Jido, and E. Oset, The KD, eta D_s interaction in finite volume and the D_{s^*0}(2317) resonance, arXiv:1109.0396 [hep-lat], 2011.
R. J. Mathar, Hierarchical Subdivision of the Simple Cubic Lattice, arXiv:1309.3705 [math.MG], 2013.
S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
J. L. Mordell, The Representation Of Integers By Three Positive Squares, Michigan Math. J. 7(3): 289-290 (1960).
Eric T. Mortenson, A Kronecker-type identity and the representations of a number as a sum of three squares, arXiv:1702.01627 [math.NT], 2017.
G. Nebe and N. J. A. Sloane, Home page for this lattice
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Theta Series
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to sums of squares

Row d=3 of A122141 and of A319574, 3rd column of A286815.
Cf. A074590 (primitive solutions), A117609 (partial sums), A004215 (positions of zeros).
Analog for 4 squares: A000118.
x^2+y^2+k*z^2: A005875, A014455, A034933, A169783, A169784.
Cf. A000164, A004013, A004015, A008443, A045828, A045831, A045834, A213384.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

# JacobiTheta3 is defined in A000122.
A005875List(len) = JacobiTheta3(len, 3)
A005875List(75) |> println # Peter Luschny, Mar 12 2018

Basis( ModularForms( Gamma1(4), 3/2), 75) [1]; /* Michael Somos, Jun 25 2014 */

(sum(x^(m^2),m=-10..10))^3; seq(coeff(%,x,n), n=0..50);
Alternative:
A005875list := proc(len) series(JacobiTheta3(0, x)^3, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A005875list(75); # Peter Luschny, Oct 02 2018

SquaresR[3,Range[0,80]] (* Harvey P. Dale, Jul 21 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3, {q, 0, n}]; (* Michael Somos, Jun 25 2014 *)
a[ n_] := Length @ FindInstance[ n == x^2 + y^2 + z^2, {x, y, z}, Integers, 10^9]; (* Michael Somos, May 21 2015 *)
QP = QPochhammer; CoefficientList[(QP[q^2]^5/(QP[q]*QP[q^4])^2)^3 + O[q]^80, q] (* Jean-François Alcover, Nov 24 2015 *)

{a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^3, n))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2)^3, n))}; /* Michael Somos, Jun 03 2012 */

{a(n) = my(G); if( n<0, 0, G = [ 1, 0, 0; 0, 1, 0; 0, 0, 1]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, May 21 2015 */

# uses Python code for A004018
from math import isqrt
def A005875(n): return A004018(n)+(sum(A004018(n-k**2) for k in range(1,isqrt(n)+1))<<1) # Chai Wah Wu, Jun 21 2024

Q = DiagonalQuadraticForm(ZZ, [1]*3)
Q.representation_number_list(75) # Peter Luschny, Jun 20 2014

A number n is representable as the sum of 3 squares iff n is not of the form 4^a (8k+7) (cf. A000378).
There is a classical formula (essentially due to Gauss):
For sums of 3 squares r_3(n): write (uniquely) -n=D(2^vf)^2, with D<0 fundamental discriminant, f odd, v>=-1. Then r_3(n) = 12L((D/.),0)(1-(D/2)) Sum_{d | f} mu(d)(D/d)sigma(f/d).
Here mu is the Moebius function, (D/2) and (D/d) are Kronecker-Legendre symbols, sigma is the sum of divisors function, L((D/.),0)=h(D)/(w(D)/2) is the value at 0 of the L function of the quadratic character (D/.), equal to the class number h(D) divided by 2 or 3 in the special cases D=-4 and -3. - Henri Cohen (Henri.Cohen(AT)math.u-bordeaux1.fr), May 12 2010
a(n) = 3*T(n) if n == 1,2,5,6 mod 8, = 2*T(n) if n == 3 mod 8, = 0 if n == 7 mod 8 and = a(n/4) if n == 0 mod 4, where T(n) = A117726(n). [Moreno-Wagstaff].
"If 12E(n) is the number of representations of n as a sum of three squares, then E(n) = 2F(n) - G(n) where G(n) = number of classes of determinant -n, F(n) = number of uneven classes." - Dickson, quoting Kronecker. [Cf. A117726.]
a(n) = Sum_{d^2|n} b(n/d^2), where b() = A074590() gives the number of primitive solutions.
Expansion of phi(q)^3 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Oct 25 2006.
Euler transform of period 4 sequence [ 6, -9, 6, -3, ...]. - Michael Somos, Oct 25 2006
G.f.: (Sum_{k in Z} x^(k^2))^3.
a(8*n + 7) = 0. a(4*n) = a(n).
a(n) = A004015(2*n) = A014455(2*n) = A004013(4*n) = A169783(4*n). a(4*n + 1) = 6 * A045834(n). a(8*n + 3) = 8 * A008443(n). a(8*n + 5) = 24 * A045831(n). - Michael Somos, Jun 03 2012
a(4*n + 2) = 12 * A045828(n). - Michael Somos, Sep 03 2014
a(n) = (-1)^n * A213384(n). - Michael Somos, May 21 2015
a(n) = (6/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017
a(n) = A004018(n) + 2*Sum_{k=1..floor(sqrt(n))} A004018(n - k^2). - Daniel Suteu, Aug 27 2021
Convolution cube of A000122. Convolution of A004018 and A000122. - R. J. Mathar, Aug 03 2025

More terms from James Sellers, Aug 22 2000

A185576 Decimal expansion of Born's basic potential Pi_0.

Original entry on oeis.org

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

A090734 Decimal expansion of 4th Madelung constant (negated).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jan 18 2004

			-1.83939908404504706624730547956723047642278359481773...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 77.

Cf. A059750, A088537, A085469.

RealDigits[8*(5 - 3*Sqrt[2])*Zeta[1/2]*Zeta[-1/2], 10, 120][[1]] (* Amiram Eldar, May 26 2023 *)

8*(5-3*sqrt(2))*zeta(1/2)*zeta(-1/2) \\ Charles R Greathouse IV, Jun 07 2016

M_4 = -8*(5-3*sqrt(2))*zeta(1/2)*zeta(-1/2) = -8 *(A157122-6) * A059750 * A211113.

A247040 Decimal expansion of M_6, the 6th Madelung constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 10 2014

			-1.9655570390090782813123135557351853678689767284464645117...

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

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's MathWorld, Dirichlet Beta Function
Eric Weisstein's MathWorld, Madelung Constants

Cf. A059750, A088537, A085469, A090734.

beta[x_] := (Zeta[x, 1/4] - Zeta[x, 3/4])/4^x; M6 = (3/Pi^2)*(4*(Sqrt[2]-1)*Zeta[1/2]*beta[5/2] - (4*Sqrt[2]-1)*Zeta[5/2]*beta[1/2]); RealDigits[M6, 10, 104][[1]]

th4(x)=1+2*sumalt(n=1, (-1)^n*x^n^2)
intnum(x=0, [oo, 1], (th4(exp(-x))^6-1)/sqrt(Pi*x)) \\ Charles R Greathouse IV, Jun 07 2016

th4(x)=1+2*sumalt(n=1, (-1)^n*x^n^2)
intnum(x=0, [oo, 1], (th4(exp(-x))^6-1)/sqrt(Pi*x)) \\ Charles R Greathouse IV, Jun 06 2016

M6 = (3/Pi^2)*(4*(sqrt(2)-1)*zeta(1/2)*beta(5/2) - (4*sqrt(2)-1)*zeta(5/2)*beta(1/2)), where beta is Dirichlet's "beta" function.

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

Coulomb self-energy of a system of electric charges is the total electrostatic potential energy of interaction between charge elements.

For a uniformly charged three-dimensional cube with a total charge Q and an edge length L it is equal to c * k*Q^2/L, where k is the Coulomb constant (A182999) and c is this constant.

cp's OEIS Frontend

A108778 Continued fraction expansion of the Madelung constant for the NaCl structure (A085469).

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A096193 Engel expansion of the Madelung constant A085469.

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A005875 Theta series of simple cubic lattice; also number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed).

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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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A090734 Decimal expansion of 4th Madelung constant (negated).

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A247040 Decimal expansion of M_6, the 6th Madelung constant.

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