cp's OEIS Frontend

A085469 Decimal expansion of Madelung constant (negated) for NaCl structure.

%I A085469 #92 Jul 01 2025 16:16:31
%S A085469 1,7,4,7,5,6,4,5,9,4,6,3,3,1,8,2,1,9,0,6,3,6,2,1,2,0,3,5,5,4,4,3,9,7,
%T A085469 4,0,3,4,8,5,1,6,1,4,3,6,6,2,4,7,4,1,7,5,8,1,5,2,8,2,5,3,5,0,7,6,5,0,
%U A085469 4,0,6,2,3,5,3,2,7,6,1,1,7,9,8,9,0,7,5,8,3,6,2,6,9,4,6,0,7,8,8,9,9,3
%N A085469 Decimal expansion of Madelung constant (negated) for NaCl structure.
%C A085469 This is the electrostatic potential at the origin produced by unit charges of sign (-1)^(i+j+k) at all nonzero lattice points (i,j,k).
%C A085469 The NaCl structure consists of two interpenetrating face-centered cubic lattices of ions with charges +1 and -1, together occupying all the sites of the simple cubic lattice. - _Andrey Zabolotskiy_, Oct 21 2019
%C A085469 Named after the German physicist Erwin Madelung (1881-1972). - _Amiram Eldar_, Apr 02 2022
%D A085469 Richard E. Crandall, Topics in Advanced Scientific Computation, Springer, Telos books, 1996, pages 73-79.
%D A085469 Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 76.
%D A085469 Sadri Hassani, Mathematical Methods Using Mathematica: For Students of Physics and Related Fields, Springer, NY, page 60.
%H A085469 Jean-François Alcover, <a href="/A085469/b085469.txt">Table of n, a(n) for n = 1..2003</a> [There were errors in the previous b-file, which had 1847 terms contributed by Harry J. Smith.]
%H A085469 David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor and Eric W. Weisstein, <a href="http://www.jstor.org/stable/27641975">Ten Problems in Experimental Mathematics</a>, Amer. Math. Monthly, Vol. 113, No. 6 (2006), pp. 481-509.
%H A085469 R. E. Crandall and J. P. Buhler, <a href="http://dx.doi.org/10.1088/0305-4470/20/16/024">Elementary function expansions for Madelung constants</a>, J. Phys. A: Math. Gen., Vol. 20, No. 16 (1987), pp. 5497-5510.
%H A085469 R. E. Crandall and J. P. Buhler, <a href="http://dx.doi.org/10.1088/0305-4470/20/9/016">The potential within a crystal lattice</a>, J. Phys. A: Math. Gen., Vol. 20, No. 9 (1987), pp. 2279-2292.
%H A085469 E. R. Fuller, Jr. and E. R. Naimon, <a href="http://dx.doi.org/10.1103/PhysRevB.6.3609">Electrostatic Contributions to the Brugger-Type Elastic Constants</a>, Phys. Rev. B, Vol. 6, No. 10 (1971), pp. 3609-3620.
%H A085469 Leslie Glasser, <a href="http://dx.doi.org/10.1021/ic2023852">Solid-State Energetics and Electrostatics: Madelung Constants and Madelung Energies</a>, Inorg. Chem., Vol. 51, No. 4 (2012), pp. 2420-2424; DOI: 10.1021/ic2023852.
%H A085469 André Hautot, <a href="http://dx.doi.org/10.1088/0305-4470/8/6/004">New applications of Poisson's summation formula</a>, J. of Phys. A, Vol. 8, No. 6 (1975) pp. 853-862.
%H A085469 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/madelung.txt">Madelung constant</a>.
%H A085469 Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap55.html">The Levy constant</a>.
%H A085469 Nicolas Tavernier, Gian Luigi Bendazzoli, Véronique Brumas, Stefano Evangelisti, and J. A. Berger, <a href="https://arxiv.org/abs/2006.01259">Clifford boundary conditions: a simple direct-sum evaluation of Madelung constants</a>, arXiv:2006.01259 [physics.comp-ph], 2020.
%H A085469 Nicolas Tavernier, Gian Luigi Bendazzoli, Véronique Brumas, Stefano Evangelisti, and J. Arjan Berger, <a href="https://arxiv.org/abs/2107.04686">Clifford Boundary Conditions for Periodic Systems: the Madelung Constant of Cubic Crystals in 1, 2 and 3 Dimensions</a>, arXiv:2107.04686 [cond-mat.mtrl-sci], 2021.
%H A085469 Sandeep Tyagi, <a href="http://dx.doi.org/10.1143/PTP.114.517">New series representation of the Madelung constant</a>, Prog. Theor. Phys., Vol. 114, No. 3 (2005), pp. 517-521.
%H A085469 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BensonsFormula.html">Benson's Formula</a>.
%H A085469 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MadelungConstants.html">Madelung Constants</a>.
%H A085469 Wikipedia, <a href="http://en.wikipedia.org/wiki/Madelung_constant">Madelung constant</a>.
%H A085469 <a href="/index/Fa#fcc">Index entries for sequences related to f.c.c. lattice</a>.
%F A085469 Sum_{i, j, k not all 0} (-1)^(i+j+k)/sqrt(i^2+j^2+k^2).
%e A085469 -1.7475645946331821906362120355443974034851614366247417581528253507...
%t A085469 RealDigits[ 12Pi*Sum[ Sech[Pi/2*Sqrt[(2j + 1)^2 + (2k + 1)^2]]^2, {j, 0, 40}, {k, 0, 40}], 10, 111][[1]] (* _Robert G. Wilson v_, Jul 12 2005 *)
%t A085469 RealDigits[Quiet[12 Pi (Sech[Pi/Sqrt[2]]^2 + NSum[Sum[Sech[Pi Norm[2 v + 1]/2]^2, {v, FrobeniusSolve[{1, 1}, Round[m]]}, Method -> "Procedural"], {m, 1, Infinity}, Compiled -> False, Method -> "WynnEpsilon", NSumTerms -> 33, WorkingPrecision -> 100])]][[1]] (* _Jan Mangaldan_, Jun 25 2020 *)
%t A085469 digits = 1800; m0 = 800; dm = 10; dd = 10; Clear[f, g];
%t A085469 g[j_, k_] := g[j, k] = 12 Pi Sech[(Pi/2) Sqrt[(2j + 1)^2 + (2k + 1)^2]]^2 // N[#, digits + dd]&;
%t A085469 f[m_] := f[m] =  Sum[g[j, k], {j, 0, m}, {k, 0, m}];
%t A085469 f[m = m0]; f[m += dm];
%t A085469 While[Abs[f[m] - f[m - dm]] > 10^(-digits - dd), Print[m]; m += dm];
%t A085469 A085469 = f[m];
%t A085469 RealDigits[A085469, 10, digits][[1]] (* _Jean-François Alcover_, May 08 2021, after _Robert G. Wilson v_ *)
%o A085469 (PARI) Madelung()=my(c=Pi/2,d=asech(2^-default(realbitprecision))\/c+1); sum(j=0,d, sum(k=0,d, sech(c*sqrt((2*j+1)^2+(2*k+1)^2))),0.)*12*Pi \\ _Charles R Greathouse IV_, Feb 07 2025
%Y A085469 Cf. A004015, A005875, A108778 (continued fraction).
%K A085469 nonn,cons
%O A085469 1,2
%A A085469 _Eric W. Weisstein_, Jul 01 2003
%E A085469 Entry revised by _N. J. A. Sloane_, Apr 12 2004
%E A085469 Definition corrected by _Leslie Glasser_, Jan 24 2011
%E A085469 Definition corrected by _Andrey Zabolotskiy_, Oct 21 2019