cp's OEIS Frontend

A019442 Numbers m such that a Hadamard matrix of order m exists.

%I A019442 #86 Feb 16 2025 08:32:33
%S A019442 1,2,4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,
%T A019442 92,96,100,104,108,112,116,120,124,128,132,136,140,144,148,152,156,
%U A019442 160,164,168,172,176,180,184,188,192,196,200,204,208,212,216,220,224,228,232,236,240
%N A019442 Numbers m such that a Hadamard matrix of order m exists.
%C A019442 It is conjectured that this sequence consists of 1, 2 and all multiples of 4.
%C A019442 Already in 1992 Hadamard matrices were known of all orders 4t up through 424.
%C A019442 The old entry with this sequence number was a duplicate of A007740.
%C A019442 Integers m such that a simplex of dimension m - 1 can be inscribed in a hypercube of dimension m - 1. - _Violeta Hernández Palacios_, Oct 23 2020
%C A019442 Integers m such that an orthoplex of dimension m can be inscribed in a hypercube of dimension m. - _Violeta Hernández Palacios_, Dec 05 2020
%C A019442 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 (see comment in A007299). - _Bernard Schott_, Apr 25 2022; Mar 03 2023
%D A019442 J. Hadamard, Résolution d'une question relative aux déterminants. Bull. des Sciences Math. (2), 17, 1893, pp. 240-246.
%D A019442 M. Hall, Jr., Hadamard matrices of order 16. Research Summary No. 36-10, Jet Propulsion Lab., Pasadena, CA, Vol. 1, 1961, pp. 21-26.
%D A019442 M. Hall, Jr., Hadamard matrices of order 20. Technical Report 32-761, Jet Propulsion Lab., Pasadena, CA, 1965.
%D A019442 M. Hall, Jr., Combinatorial Theory. 2nd edn. New York: Wiley, 1986.
%D A019442 S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, Springer-Verlag, NY, 1999, Chapter 7.
%D A019442 Jennifer Seberry and Mieko Yamada, Hadamard matrices, sequences and block designs, in Dinitz and Stinson, eds., Contemporary design theory, pp. 431-560, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1992.
%D A019442 W. D. Wallis, Anne Penfold Street, and Jennifer Seberry Wallis; Combinatorics: Room squares, sum-free sets, Hadamard matrices. Lecture Notes in Mathematics, Vol. 292. Springer-Verlag, Berlin-New York, 1972. iv+508 pp.
%H A019442 J. Adams, P. Zvengrowski, and P. Laird, <a href="https://doi.org/10.1016/S0723-0869(03)80037-3">Vertex embeddings of regular polytopes</a>, Expositiones Mathematicae, (4), 21, 2003, pp. 339-353.
%H A019442 D. Z. Djokovic, <a href="http://dx.doi.org/10.1017/S0004972700030185">Construction of some new Hadamard matrices</a>, Bull. Austral. Math. Soc., Volume 45, Issue 2, April 1992, pp. 327-332.
%H A019442 D. Z. Djokovic, <a href="http://ajc.maths.uq.edu.au/pdf/10/ajc-v10-p259.pdf">Five new orders for Hadamard matrices of skew type</a>. Australas. J. Combin., Vol. 10, 1994.
%H A019442 D. Z. Djokovic, <a href="http://dx.doi.org/10.1007/BF01212983">Two Hadamard matrices of order 956 of Goethals-Seidel type</a>, Combinatorica, 1994, 14, 375-377.
%H A019442 J. Hadamard, <a href="http://mathscinet.ru/files/HadamardTranslation.pdf">Résolution d'une question relative aux déterminants</a>, Bull. des Sciences Math. (2), 17, 1893, 240-246. English translation.
%H A019442 Hiroshi Kimura, <a href="http://dx.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 A019442 Hiroshi Kimura, <a href="http://dx.doi.org/10.1007/BF01788676">New Hadamard matrix of order 24</a>, Graphs Combin., 1989, 5, 235-242.
%H A019442 Hiroshi Kimura, <a href="http://dx.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 A019442 Hiroshi Kimura, <a href="http://dx.doi.org/10.1016/0012-365X(94)90024-8">Classification of Hadamard matrices of order 28</a>, Discrete Math., 1994, 133, 171-180.
%H A019442 W. P. Orrick, <a href="http://arxiv.org/abs/math.CO/0507515">Switching operations for Hadamard matrices</a>, arXiv:math/0507515 [math.CO], 2005-2007.
%H A019442 R. E. A. C. Paley, <a href="http://dx.doi.org/10.1002/sapm1933121311">On orthogonal matrices</a>, J. Math. Phys., 12, 311-320.
%H A019442 R. L. Plackett and J. P. Burman, <a href="http://www.jstor.org/stable/2332195">The design of optimum multifactorial experiments</a>, Biometrika, 1946, 33, 305-325.
%H A019442 K. Sawade, <a href="http://dx.doi.org/10.1007/BF02582942">A Hadamard matrix of order 268</a>, Graphs Combin., 1985, 1, 185-187.
%H A019442 Jennifer Seberry and Mieko Yamada, <a href="https://ro.uow.edu.au/infopapers/1070/">Hadamard matrices, sequences and block designs</a>, in Dinitz and Stinson, eds., Contemporary design theory, pp. 431-560, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1992.
%H A019442 N. J. A. Sloane, <a href="http://neilsloane.com/hadamard/">Tables of Hadamard matrices</a>.
%H A019442 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 A019442 Edward Spence, <a href="http://dx.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 A019442 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HadamardMatrix.html">Hadamard Matrix</a>.
%H A019442 J. Williamson, <a href="http://dx.doi.org/10.1215/S0012-7094-44-01108-7">Hadamard's determinant theorem and the sum of four squares</a>, Duke Math. J., 1994, 11, 65-81.
%H A019442 <a href="/index/Ha#Hadamard">Index entries for sequences related to Hadamard matrices</a>
%F A019442 Conjectured g.f.: (2*x^3 + x^2 + 1)/(x - 1)^2. - _Jean-François Alcover_, Oct 03 2016
%Y A019442 Cf. A007299, A036297, A016742.
%K A019442 nonn,nice
%O A019442 1,2
%A A019442 _N. J. A. Sloane_, Oct 16 2008