%I M3736 #84 Oct 19 2024 15:57:32

%S 1,1,1,1,5,3,60,487,13710027

%N Number of Hadamard matrices of order 4n.

%C More precisely, number of inequivalent Hadamard matrices of order n if two matrices are considered equivalent if one can be obtained from the other by permuting rows, permuting columns and multiplying rows or columns by -1.

%C The Hadamard conjecture is that a(n) > 0 for all n >= 0. - _Charles R Greathouse IV_, Oct 08 2012

%C From _Bernard Schott_, Apr 24 2022: (Start)

%C A brief historical overview based on the article "La conjecture de Hadamard" (see link):

%C 1893 - J. Hadamard proposes his conjecture: a Hadamard matrix of order 4k exists for every positive integer k (see link).

%C As of 2000, there were five multiples of 4 less than or equal to 1000 for which no Hadamard matrix of that order was known: 428, 668, 716, 764 and 892.

%C 2005 - Hadi Kharaghani and Behruz Tayfeh-Rezaie publish their construction of a Hadamard matrix of order 428 (see link).

%C 2007 - D. Z. Djoković publishes "Hadamard matrices of order 764 exist" and constructs 2 such matrices (see link).

%C As of today, there remain 12 multiples of 4 less than or equal to 2000 for which no Hadamard matrix of that order is known: 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, and 1964. (End)

%C By private email, _Felix A. Pahl_ informs that a Hadamard matrix of order 1004 was constructed in 2013 (see link Djoković, Golubitsky, Kotsireas); so 1004 is deleted from the last comment. - _Bernard Schott_, Jan 29 2023

%D J. Hadamard, Résolution d'une question relative aux déterminants, Bull. des Sciences Math. 2 (1893), 240-246.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D S. Wolfram, A New Kind of Science. Champaign, IL: Wolfram Media, p. 1073, 2002.

%H V. Alvarez, J. A. Armario, M. D. Frau and F. Gudlel, <a href="https://doi.org/10.1016/j.laa.2011.05.018">The maximal determinant of cocyclic (-1, 1)-matrices over D_{2t}</a>, Linear Algebra and its Applications, 2011.

%H F. J. Aragon Artacho, J. M. Borwein, and M. K. Tam, <a href="https://carmamaths.org/resources/jon/DR_MatrixCompletion.pdf">Douglas-Rachford Feasibility Methods for Matrix Completion Problems</a>, March 2014.

%H Dragomir Z. Djoković, <a href="https://arxiv.org/abs/math/0703312">Hadamard matrices of order 764 exist</a>, arXiv:math/0703312 [math.CO], 2007.

%H Dragomir Z. Djoković, Oleg Golubitsky, and Ilias S. Kotsireas, <a href="https://arxiv.org/abs/1301.3671">Some new orders of Hadamard and skew-Hadamard matrices</a>, arXiv:1301.3671 [math.CO], 2013.

%H Shalom Eliahou, <a href="http://images-archive.math.cnrs.fr/La-conjecture-de-Hadamard-I.html">La conjecture de Hadamard (I)</a>, Images des Mathématiques, CNRS, 2020.

%H Shalom Eliahou, <a href="http://images-archive.math.cnrs.fr/La-conjecture-de-Hadamard-II">La conjecture de Hadamard (II)</a>, Images des Mathématiques, CNRS, 2020.

%H Jacques Salomon Hadamard, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k30724/f1502n2.capture">Sur le module maximum que puisse atteindre un déterminant</a>, C. R. Acad. Sci. Paris 116 (1893) 1500-1501 (link from Gallica).

%H Hadi Kharaghani, <a href="http://www.cs.uleth.ca/~hadi/">Home Page</a>

%H Hadi Kharaghani and B. Tayfeh-Rezaie, <a href="http://math.ipm.ac.ir/tayfeh-r/Hadamard32.htm">Hadamard matrices of order 32</a>, J. Combin. Designs 21 (2013) no. 5, 212-221. [<a href="http://dx.doi.org/10.1002/jcd.21323">DOI</a>]

%H Hadi Kharaghani and B. Tayfeh-Rezaie, <a href="https://doi.org/10.1002/jcd.20043">A Hadamard matrix of order 428</a>, Journal of Combinatorial Designs, Volume 13, Issue 6, November 2005, pp. 435-440 (First published: 13 December 2004).

%H H. Kharaghani and B. Tayfeh-Rezaie, <a href="https://www.cs.uleth.ca/~hadi/research/h32/h32-classification-fv.pdf">On the classification of Hadamard matrices of order 32</a>, J. Combin. Des., 18 (2010), 328-336.

%H H. Kharaghani and B. Tayfeh-Rezaie, <a href="http://math.ipm.ac.ir/~tayfeh-r/papersandpreprints/H32typetwo.pdf">Hadamard matrices of order 32</a>, 2012.

%H H. Kimura, <a href="https://doi.org/10.1016/0097-3165(86)90027-0">Hadamard matrices of order 28 with automorphism groups of order two</a>, J. Combin. Theory (1986), A 43, 98-102.

%H H. Kimura, <a href="https://doi.org/10.1007/BF01788676">New Hadamard matrix of order 24</a>, Graphs Combin. (1989), 5, 235-242.

%H H. Kimura, <a href="https://doi.org/10.1016/0012-365X(94)90117-1">Classification of Hadamard matrices of order 28 with Hall sets</a>, Discrete Math. (1994), 128, 257-268.

%H H. Kimura, <a href="https://doi.org/10.1016/0012-365X(94)90024-8">Classification of Hadamard matrices of order 28</a>, Discrete Math. (1994), 133, 171-180.

%H W. P. Orrick, <a href="https://arxiv.org/abs/math/0507515">Switching operations for Hadamard matrices</a>, arXiv:math/0507515 [math.CO], 2005-2007. (Gives lower bounds for a(8) and a(9))

%H N. J. A. Sloane, <a href="http://neilsloane.com/hadamard/">Tables of Hadamard matrices</a>

%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).

%H Warren D. Smith, <a href="/A007299/a007299.c.txt">C program for generating Hadamard matrices of various orders</a>

%H Edward Spence, <a href="https://doi.org/10.1016/0012-365X(93)E0169-5">Classification of Hadamard matrices of order 24 and 28</a>, Discrete Math. 140 (1995), no. 1-3, 185-243.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HadamardMatrix.html">Hadamard Matrix</a>

%H <a href="/index/Ha#Hadamard">Index entries for sequences related to Hadamard matrices</a>

%Y Cf. A019442, A096201, A036297, A048615, A048616, A003432, A048885.

%K hard,nonn,nice

%O 0,5

%A _N. J. A. Sloane_

%E a(8) from the H. Kharaghani and B. Tayfeh-Rezaie paper. - _N. J. A. Sloane_, Feb 11 2012