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%I
%S 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 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 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 Decimal expansion of Madelung constant (negated) for NaCl structure.
%C 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 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
%D Richard E. Crandall, Topics in Advanced Scientific Computation, Springer, Telos books, 1996, pages 73-79.
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 76.
%D Sadri Hassani, Mathematical Methods Using Mathematica: For Students of Physics and Related Fields, Springer, NY, page 60.
%H Harry J. Smith, <a href="/A085469/b085469.txt">Table of n, a(n) for n = 1..1847</a>
%H D. H. Bailey, J. M. Borwein, V. Kapoor and E. Weisstein, <a href="http://www.jstor.org/stable/27641975">Ten Problems in Experimental Mathematics</a>, Amer. Math. Monthly 113 (6) (2006), 481-509.
%H 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. 20 (1987) no. 16, 5497-5510.
%H 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. 20 (1987) no. 9, 2279-2292.
%H 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 6 (1971) no. 10, 3609-3620.
%H Leslie Glasser, <a href="http://dx.doi.org/10.1021/ic2023852">Solid-State Energetics and Electrostatics: Madelung Constants and Madelung Energies</a>, Inorg. Chem., 2012, 51 (4), 2420-2424; DOI: 10.1021/ic2023852.
%H 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 #6, 1975 pp. 853-862.
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/madelung.txt">Madelung constant</a>
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap55.html">The Levy constant</a>
%H Nicolas Tavernier, Gian Luigi Bendazzoli, Véronique Brumas, Stefano Evangelisti, 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 Sandeep Tyagi, <a href="http://dx.doi.org/10.1143/PTP.114.517">New series representation of the Madelung constant</a>, Prog. Theor. Phys. 114 (2005) No. 3, 517-521.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BensonsFormula.html">Benson's Formula</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MadelungConstants.html">Madelung Constants</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Madelung_constant">Madelung constant</a>
%H <a href="/index/Fa#fcc">Index entries for sequences related to f.c.c. lattice</a>
%F Sum_{i, j, k not all 0} (-1)^(i+j+k)/sqrt(i^2+j^2+k^2).
%e -1.7475645946331821906362120355443974034851614366247417581528253507...
%t 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 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 *)
%Y Cf. A004015, A005875, A108778 (continued fraction).
%K nonn,cons
%O 1,2
%A _Eric W. Weisstein_, Jul 01 2003
%E Entry revised by _N. J. A. Sloane_, Apr 12, 2004
%E Definition corrected by _Leslie Glasser_, Jan 24 2011
%E Definition corrected by _Andrey Zabolotskiy_, Oct 21 2019
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