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Search: id:A085469
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| A085469 |
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Decimal expansion of Madelung constant (negated) for simple cubic lattice. |
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+0
4
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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
(list; cons; graph; listen)
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OFFSET
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1,2
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COMMENT
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This is the electrostatic potential at the origin produced by unit charges at all nonzero lattice points.
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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
Andre Hautot, New applications of Poisson's summation formula, J of Phys, A vol. 8 #6, 1975 pp. 853-862.
Sadri Hassani, Mathematical Methods Using Mahematica: 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
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
Simon Plouffe, Madelung constant
S. Plouffe, The Levy constant
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
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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]] (from Robert G. Wilson v (rgwv(at)rgwv.com), Jul 12 2005)
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PROGRAM
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Contribution from Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 15 2009: (Start)
(PARI) { default(realprecision, 1848); Madelung=-1.7475645946331821906362120355443974034851614366247417581\
5282535076504062353276117989075836269460788993083258153875371059328\
2029944183828013036933002156599363282376607172297568659238037167203\
8104106034214556064382777786832173132243697558773426250474787821285\
0860567916681675739924476841297036782518576281093713133720767071931\
9742497158115723096992309669273949657781107222671520547409011506891\
5716583082820050184892117803134673122964985828828184357133159143170\
0549563253348875363026704256274869484380028002592700268475574364975\
5049224613623992040015750630397214664811151237364010295066011939046\
7194373312530445102911514639759331918047977946099333746429426562908\
9693447792968854190440791425583272199718409067468023761538935445655\
0360273028544084934430280626704418241200439741867661772447563953444\
2306853849527943580751895490309305073843954464206438717926390780392\
0744282097957917736992304082214374645668043105692663197550459224432\
4807489408062474936107093630914922436898693314090379682324079004628\
4485812201497496519179081118204181820099174737652482957295684972796\
0238432587361742516304301213253823307779481444598420343673216212916\
2257903116101353527417534916776824438057139382407124829068734888254\
4125452570636795213611364128355999680725389089412120267587472631283\
4068262537606780890814341434286117388903537106985249308735988759660\
2363184706607516442145683455586623676605437844742512217481810852938\
2986359330038858941133489312302314299432802377837641811377123806642\
4808681158556451188180538368104068069306152452657895325624057864079\
5650016172016637963148270800941668173022273735813292616204948281860\
4313652035571121839219375759423906825690022886650812437760991098879\
3853419448765682333487553786009201932911164565934512096197118655146\
2827591256212354525900502281084479644187471007265894559896882147628\
71074726392395758480653025376082029590142679877534; x=-Madelung; for (n=1, 1847, d=floor(x); x=(x-d)*10; write("b085469.txt", n, " ", d)); } (End)
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CROSSREFS
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Cf. A005875.
Cf. A108778 = Continued fraction. [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 13 2009]
Sequence in context: A160575 A153586 A153186 this_sequence A050996 A085541 A133055
Adjacent sequences: A085466 A085467 A085468 this_sequence A085470 A085471 A085472
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KEYWORD
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nonn,cons
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AUTHOR
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Eric Weisstein (eric(AT)weisstein.com), Jul 01, 2003
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EXTENSIONS
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Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Apr 12, 2004
Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 19 2009
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