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A085469
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Decimal expansion of Madelung constant (negated) for NaCl structure.
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20
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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, 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, 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
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OFFSET
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1,2
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COMMENTS
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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).
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
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REFERENCES
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Richard E. Crandall, Topics in Advanced Scientific Computation, Springer, Telos books, 1996, pages 73-79.
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 76.
Sadri Hassani, Mathematical Methods Using Mathematica: For Students of Physics and Related Fields, Springer, NY, page 60.
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LINKS
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Harry J. Smith, Table of n, a(n) for n = 1..1847
D. H. Bailey, J. M. Borwein, V. Kapoor and E. Weisstein, Ten Problems in Experimental Mathematics, Amer. Math. Monthly 113 (6) (2006), 481-509.
R. E. Crandall and J. P. Buhler, Elementary function expansions for Madelung constants, J. Phys. A: Math. Gen. 20 (1987) no. 16, 5497-5510.
R. E. Crandall and J. P. Buhler, The potential within a crystal lattice, J. Phys. A: Math. Gen. 20 (1987) no. 9, 2279-2292.
E. R. Fuller Jr and E. R. Naimon, Electrostatic Contributions to the Brugger-Type Elastic Constants, Phys. Rev. B 6 (1971) no. 10, 3609-3620.
Leslie Glasser, Solid-State Energetics and Electrostatics: Madelung Constants and Madelung Energies, Inorg. Chem., 2012, 51 (4), 2420-2424; DOI: 10.1021/ic2023852.
André Hautot, New applications of Poisson's summation formula, J of Phys, A vol. 8 #6, 1975 pp. 853-862.
Simon Plouffe, Madelung constant
Simon Plouffe, The Levy constant
Nicolas Tavernier, Gian Luigi Bendazzoli, Véronique Brumas, Stefano Evangelisti, J. A. Berger, Clifford boundary conditions: a simple direct-sum evaluation of Madelung constants, arXiv:2006.01259 [physics.comp-ph], 2020.
Sandeep Tyagi, New series representation of the Madelung constant, Prog. Theor. Phys. 114 (2005) No. 3, 517-521.
Eric Weisstein's World of Mathematics, Benson's Formula
Eric Weisstein's World of Mathematics, Madelung Constants
Wikipedia, Madelung constant
Index entries for sequences related to f.c.c. lattice
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FORMULA
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Sum_{i, j, k not all 0} (-1)^(i+j+k)/sqrt(i^2+j^2+k^2).
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EXAMPLE
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-1.7475645946331821906362120355443974034851614366247417581528253507...
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MATHEMATICA
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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 *)
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 *)
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CROSSREFS
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Cf. A004015, A005875, A108778 (continued fraction).
Sequence in context: A153586 A319883 A153186 * A050996 A085541 A133055
Adjacent sequences: A085466 A085467 A085468 * A085470 A085471 A085472
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KEYWORD
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nonn,cons
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AUTHOR
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Eric W. Weisstein, Jul 01 2003
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EXTENSIONS
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Entry revised by N. J. A. Sloane, Apr 12, 2004
Definition corrected by Leslie Glasser, Jan 24 2011
Definition corrected by Andrey Zabolotskiy, Oct 21 2019
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STATUS
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approved
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