A048616 -id:A048616 - cp's OEIS frontend

A007299 Number of Hadamard matrices of order 4n.

1, 1, 1, 1, 5, 3, 60, 487, 13710027

Offset: 0

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.
The Hadamard conjecture is that a(n) > 0 for all n >= 0. - Charles R Greathouse IV, Oct 08 2012
From Bernard Schott, Apr 24 2022: (Start)
A brief historical overview based on the article "La conjecture de Hadamard" (see link):
1893 - J. Hadamard proposes his conjecture: a Hadamard matrix of order 4k exists for every positive integer k (see link).
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.
2005 - Hadi Kharaghani and Behruz Tayfeh-Rezaie publish their construction of a Hadamard matrix of order 428 (see link).
2007 - D. Z. Djoković publishes "Hadamard matrices of order 764 exist" and constructs 2 such matrices (see link).
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)
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

J. Hadamard, Résolution d'une question relative aux déterminants, Bull. des Sciences Math. 2 (1893), 240-246.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Wolfram, A New Kind of Science. Champaign, IL: Wolfram Media, p. 1073, 2002.

V. Álvarez, J. A. Armario, M. D. Frau, and F. Gudiel, The maximal determinant of cocyclic (-1, 1)-matrices over D_{2t}, Linear Algebra and its Applications, 2011.
F. J. Aragon Artacho, J. M. Borwein, and M. K. Tam, Douglas-Rachford Feasibility Methods for Matrix Completion Problems, March 2014.
Dragomir Z. Djoković, Hadamard matrices of order 764 exist, arXiv:math/0703312 [math.CO], 2007.
Dragomir Z. Djoković, Oleg Golubitsky, and Ilias S. Kotsireas, Some new orders of Hadamard and skew-Hadamard matrices, arXiv:1301.3671 [math.CO], 2013.
Shalom Eliahou, La conjecture de Hadamard (I), Images des Mathématiques, CNRS, 2020.
Shalom Eliahou, La conjecture de Hadamard (II), Images des Mathématiques, CNRS, 2020.
Jacques Salomon Hadamard, Sur le module maximum que puisse atteindre un déterminant, C. R. Acad. Sci. Paris 116 (1893) 1500-1501 (link from Gallica).
Hadi Kharaghani, Home Page
Hadi Kharaghani and B. Tayfeh-Rezaie, Hadamard matrices of order 32, J. Combin. Designs 21 (2013) no. 5, 212-221. [DOI]
Hadi Kharaghani and B. Tayfeh-Rezaie, A Hadamard matrix of order 428, Journal of Combinatorial Designs, Volume 13, Issue 6, November 2005, pp. 435-440 (First published: 13 December 2004).
H. Kharaghani and B. Tayfeh-Rezaie, On the classification of Hadamard matrices of order 32, J. Combin. Des., 18 (2010), 328-336.
H. Kharaghani and B. Tayfeh-Rezaie, Hadamard matrices of order 32, 2012.
H. Kimura, Hadamard matrices of order 28 with automorphism groups of order two, J. Combin. Theory (1986), A 43, 98-102.
H. Kimura, New Hadamard matrix of order 24, Graphs Combin. (1989), 5, 235-242.
H. Kimura, Classification of Hadamard matrices of order 28 with Hall sets, Discrete Math. (1994), 128, 257-268.
H. 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. (Gives lower bounds for a(8) and a(9))
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).
Warren D. Smith, C program for generating Hadamard matrices of various orders
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
Index entries for sequences related to Hadamard matrices

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

a(8) from the H. Kharaghani and B. Tayfeh-Rezaie paper. - N. J. A. Sloane, Feb 11 2012

A048615 Numerator of mass (Sum 1/|Aut(H)|) of Hadamard matrices of order 4n.

Original entry on oeis.org

1, 1, 1, 27, 127, 34603, 278617, 264156210586399

Offset: 1

N. J. A. Sloane, E. M. Rains

This list starts with the Hadamard matrix of order 4. The masses of the Hadamard matrices of orders 1 and 2 are 1/2 and 1/8 respectively.

			1/192, 1/21504, 1/190080, 27/573440, 127/218880, 34603/40480, 278617/2106, 264156210586399/53329920, ...

V. Alvarez, J. A. Armario, M. D. Frau and F. Gudiel, The maximal determinant of cocyclic (-1, 1)-matrices over D_{2t}, Linear Algebra and its Applications, 2011.
Index entries for sequences related to Hadamard matrices

Cf. A048616, A007299, A036297.

a(8) from Brendan McKay, Feb 11 2012

A206711 Total number of distinct Hadamard matrices of order 4n.

Original entry on oeis.org

768, 4954521600, 20251509535014912000, 88526812916367202104587059200000, 3776127947893930552689423154306445475840000000, 92624181047745713568610317051197596401168530978226831360000000, 886156947284057553944669848348035536068124589065755283423684984832000000000000

Offset: 1

Brendan McKay, Feb 11 2012 (entered by N. J. A. Sloane)

This is the total number of distinct Hadamard matrices of order 4n, ignoring all equivalences.

Brendan McKay, Table of n, a(n) for n = 1..8
J. Dong, V. Elser, G. Gyawali, K. Y. Jee, J. Kent-Dobias, A. Mandaiya, M. Renz, Y. Su, Glass phenomenology in the hard matrix model, arXiv:1912.07558 [cond-mat.stat-mech], 2019.
H. Kharaghani and B. Tayfeh-Rezaie, Hadamard matrices of order 32.
Index entries for sequences related to Hadamard matrices

Cf. A007299, A036297, A206712.

a(n) = A048615(n)/A048616(n) * (2^n * n!)^2.

a(n) = A206712(4n).

A199006 Related to number of Hadamard matrices of order 4n.

Original entry on oeis.org

192, 21504, 190080, 10838016, 16440, 823616, 74306

Offset: 1

N. J. A. Sloane, Nov 01 2011

It seems that Álvarez et al. calculate these numbers by summing the orders of Aut(H) over inequivalent Hadamard matrices H. If so, a(8) = 20643963716 from Kharaghani and Tayfeh-Rezaie's Table 3. - Andrei Zabolotskii, Jul 08 2025

V. Álvarez, J. A. Armario, M. D. Frau, and F. Gudiel, The maximal determinant of cocyclic (-1, 1)-matrices over D_{2t}, Linear Algebra and its Applications, 2011, Volume 436, Issue 4, 15 February 2012, Pages 858-873. See Table 2. The terms a(1)-a(7) are said to be computed from the data from N. J. A. Sloane's website.
Hadi Kharaghani and Behruz Tayfeh-Rezaie, Hadamard Matrices of Order 32, Journal of Combinatorial Designs, 21 (2013), 212-221; see also preprint.
N. J. A. Sloane, A Library of Hadamard Matrices.

Cf. A007299, A048615, A048616, A199005, A199007, A206711.

A206712 Total number of distinct Hadamard matrices of order n.

Original entry on oeis.org

2, 8, 0, 768, 0, 0, 0, 4954521600, 0, 0, 0, 20251509535014912000, 0, 0, 0, 88526812916367202104587059200000, 0, 0, 0, 3776127947893930552689423154306445475840000000, 0, 0, 0, 92624181047745713568610317051197596401168530978226831360000000, 0, 0, 0, 886156947284057553944669848348035536068124589065755283423684984832000000000000, 0, 0, 0

Offset: 1

Brendan McKay, Feb 11 2012 (entered by N. J. A. Sloane)

a(n) is the total number of distinct Hadamard matrices of order n, ignoring all equivalences.

Brendan McKay, Table of n, a(n) for n = 1..35
F. J. Aragon Artacho, J. M. Borwein, M. K. Tam, Douglas-Rachford Feasibility Methods for Matrix Completion Problems, March 2014.
H. Kharaghani and B. Tayfeh-Rezaie, Hadamard matrices of order 32
Index entries for sequences related to Hadamard matrices

Cf. A007299, A036297, A206711.

a(4n) = A206711(n) = A048615(n)/A048616(n) * (2^n * n!)^2.

For n>1, a(4n+1) = a(4n+2) = a(4n+3) = 0.

cp's OEIS Frontend

A007299 Number of Hadamard matrices of order 4n.

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A048615 Numerator of mass (Sum 1/|Aut(H)|) of Hadamard matrices of order 4n.

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A206711 Total number of distinct Hadamard matrices of order 4n.

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A199006 Related to number of Hadamard matrices of order 4n.

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A206712 Total number of distinct Hadamard matrices of order n.

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