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%I M5482 #116 Jun 02 2024 12:31:53
%S 1,0,1,880,275305224,17753889197660635632
%N Number of magic squares of order n composed of the numbers from 1 to n^2, counted up to rotations and reflections.
%C a(4) computed by Frenicle de Bessy (1605? - 1675), published in 1693. The article mentions the 880 squares and considers also 5*5, 6*6, 8*8, and other squares. - _Paul Curtz_, Jul 13 and Aug 12 2011
%C a(5) computed by Richard C. Schroeppel in 1973.
%C According to Pinn and Wieczerkowski, a(6) = (0.17745 +- 0.00016) * 10^20. - _R. K. Guy_, May 01 2004
%C a(6) computed by Hidetoshi Mino in 2024 - _Hidetoshi Mino_, May 31 2024
%D E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Vol. II, pp. 778-783 gives the 880 4 X 4 squares.
%D M. Gardner, Mathematical Games, Sci. Amer. Vol. 249 (No. 1, 1976), p. 118.
%D M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 216.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Ian Cameron, Adam Rogers and Peter Loly, <a href="http://www.physics.umanitoba.ca/~icamern/Poland2012/Data/Bewedlo%20Codex.pdf">"The Library of Magical Squares" -- a summary of the main results for the Shannon entropy of magic and Latin squares: isentropic clans and indexing, in celebration of George Styan's 75th</a>.
%H Bernard Frénicle de Bessy, <a href="http://babel.hathitrust.org/cgi/pt?u=1&num=423&seq=11&view=image&size=100&id=ucm.5323750390">Des carrez ou tables magiques</a>, Divers ouvrages de mathématique et de physique (1693), pp. 423-483.
%H Bernard Frénicle de Bessy, <a href="http://babel.hathitrust.org/cgi/pt?u=1&num=484&seq=9&view=image&size=100&id=ucm.5323750390">Table générale des carrez de quatre</a>, Divers ouvrages de mathématique et de physique (1693), pp. 484-503.
%H Skylar R. Croy, Jeremy A. Hansen, and Daniel J. McQuillan, <a href="https://www.aaai.org/ocs/index.php/SOCS/SOCS16/paper/viewFile/13973/13254?fbclid=IwAR3ZZ24E8vLtbrdpQ-OijrEhiUydRed1_DZP-GJk9jxczLzuuBD29XuoalM">Calculating the Number of Order-6 Magic Squares with Modular Lifting</a>, Proceedings of the Ninth International Symposium on Combinatorial Search (SoCS 2016).
%H Mahadi Hasan and Md. Masbaul Alam Polash, <a href="https://doi.org/10.1007/978-981-13-8942-9_7">An Efficient Constraint-Based Local Search for Maximizing Water Retention on Magic Squares</a>, Emerging Trends in Electrical, Communications, and Information Technologies, Lecture Notes in Electrical Engineering book series (LNEE 2019) Vol. 569, 71-79.
%H Hidetoshi Mino, <a href="https://magicsquare6.net/">The number of magic squares of order 6</a>.
%H I. Peterson, <a href="https://web.archive.org/web/20080421150630/http://www.sciencenews.org/pages/sn_arc99/10_16_99/mathland.htm">Magic Tesseracts</a>.
%H K. Pinn and C. Wieczerkowski, <a href="https://arxiv.org/abs/cond-mat/9804109">Number of magic squares from parallel tempering Monte Carlo</a>, arXiv:cond-mat/9804109 [cond-mat.stat-mech], 1998; Internat. J. Modern Phys., 9 (4) (1998) 541-546.
%H Artem Ripatti, <a href="https://arxiv.org/abs/1807.02983">On the number of semi-magic squares of order 6</a>, arXiv:1807.02983 [math.CO], 2018. See Table 1 p. 2.
%H R. Schroeppel, <a href="/A006052/a006052_2.pdf">Emails to N. J. A. Sloane, Jun. 1991</a>.
%H N. J. A. Sloane & J. R. Hendricks, <a href="/A006052/a006052_3.pdf">Correspondence, 1974</a>.
%H Walter Trump, <a href="http://www.trump.de/magic-squares/howmany.html">How many magic squares are there? - Results of historical and computer enumeration</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MagicSquare.html">Magic Square</a>.
%H <a href="/index/Mag#magic">Index entries for sequences related to magic squares</a>
%e An illustration of the unique (up to rotations and reflections) magic square of order 3:
%e +---+---+---+
%e | 2 | 7 | 6 |
%e +---+---+---+
%e | 9 | 5 | 1 |
%e +---+---+---+
%e | 4 | 3 | 8 |
%e +---+---+---+
%Y Cf. A270876, A271103, A271104.
%K nonn,hard,nice,more
%O 1,4
%A _N. J. A. Sloane_
%E Definition corrected by _Max Alekseyev_, Dec 25 2015
%E a(6) from _Hidetoshi Mino_, Jul 17 2023
%E Incorrect a(6) removed by _Hidetoshi Mino_, Sep 07 2023
%E a(6) from _Hidetoshi Mino_, May 31 2024
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