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

1, 2, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240

Offset: 1

Author

N. J. A. Sloane, Oct 16 2008

Keywords

• nonn
• nice

Comments

It is conjectured that this sequence consists of 1, 2 and all multiples of 4.

Already in 1992 Hadamard matrices were known of all orders 4t up through 424.

The old entry with this sequence number was a duplicate of A007740.

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

Integers m such that an orthoplex of dimension m can be inscribed in a hypercube of dimension m. - Violeta Hernández Palacios, Dec 05 2020

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

References

• J. Hadamard, Résolution d'une question relative aux déterminants. Bull. des Sciences Math. (2), 17, 1893, pp. 240-246.
• M. Hall, Jr., Hadamard matrices of order 16. Research Summary No. 36-10, Jet Propulsion Lab., Pasadena, CA, Vol. 1, 1961, pp. 21-26.
• M. Hall, Jr., Hadamard matrices of order 20. Technical Report 32-761, Jet Propulsion Lab., Pasadena, CA, 1965.
• M. Hall, Jr., Combinatorial Theory. 2nd edn. New York: Wiley, 1986.
• S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, Springer-Verlag, NY, 1999, Chapter 7.
• 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.
• 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.

Links

• J. Adams, P. Zvengrowski, and P. Laird, Vertex embeddings of regular polytopes, Expositiones Mathematicae, (4), 21, 2003, pp. 339-353.
• D. Z. Djokovic, Construction of some new Hadamard matrices, Bull. Austral. Math. Soc., Volume 45, Issue 2, April 1992, pp. 327-332.
• D. Z. Djokovic, Five new orders for Hadamard matrices of skew type. Australas. J. Combin., Vol. 10, 1994.
• D. Z. Djokovic, Two Hadamard matrices of order 956 of Goethals-Seidel type, Combinatorica, 1994, 14, 375-377.
• J. Hadamard, Résolution d'une question relative aux déterminants, Bull. des Sciences Math. (2), 17, 1893, 240-246. English translation.
• Hiroshi Kimura, Hadamard matrices of order 28 with automorphism groups of order two, J. Combin. Theory, 1986, A 43, 98-102.
• Hiroshi Kimura, New Hadamard matrix of order 24, Graphs Combin., 1989, 5, 235-242.
• Hiroshi Kimura, Classification of Hadamard matrices of order 28 with Hall sets, Discrete Math., 1994, 128, 257-268.
• Hiroshi Kimura, Classification of Hadamard matrices of order 28, Discrete Math., 1994, 133, 171-180.
• W. P. Orrick, Switching operations for Hadamard matrices, arXiv:math/0507515 [math.CO], 2005-2007.
• R. E. A. C. Paley, On orthogonal matrices, J. Math. Phys., 12, 311-320.
• R. L. Plackett and J. P. Burman, The design of optimum multifactorial experiments, Biometrika, 1946, 33, 305-325.
• K. Sawade, A Hadamard matrix of order 268, Graphs Combin., 1985, 1, 185-187.
• 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.
• N. J. A. Sloane, Tables of Hadamard matrices.
• N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
• Edward Spence, Classification of Hadamard matrices of order 24 and 28, Discrete Math. 140 (1995), no. 1-3, 185-243.
• Eric Weisstein's World of Mathematics, Hadamard Matrix.
• J. Williamson, Hadamard's determinant theorem and the sum of four squares, Duke Math. J., 1994, 11, 65-81.
• Index entries for sequences related to Hadamard matrices

Crossrefs

Cf. A007299, A036297, A016742.

Formula

Conjectured g.f.: (2*x^3 + x^2 + 1)/(x - 1)^2. - Jean-François Alcover, Oct 03 2016