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

1,2

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

Table of n, a(n) for n=1..62.

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

Formula

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

Crossrefs

Cf. A007299, A036297, A016742.

Sequence in context: A358308 A325326 A215459 * A048166 A360013 A335738

Adjacent sequences: A019439 A019440 A019441 * A019443 A019444 A019445

Keyword

nonn,nice

Author

N. J. A. Sloane, Oct 16 2008

Status

approved