A007299 -id:A007299 - cp's OEIS frontend

A036297 Number of Hadamard matrices of order n.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 3, 0, 0, 0, 60, 0, 0, 0, 487, 0, 0, 0, 13710027, 0, 0, 0

Offset: 0

N. J. A. Sloane

See A007299 for references.

See A007299 for links.
Index entries for sequences related to Hadamard matrices
Index entries for sequences related to maximal determinants

A007299 is the main entry for this sequence. Cf. A003432.

a(32) from the H. Kharaghani and B. Tayfeh-Rezaie paper. - N. J. A. Sloane, Feb 11 2012

A096201 Number of Hadamard matrices of order 4n when a matrix and its transpose are considered equivalent.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 36, 294, 6857010

Offset: 0

Ian Wanless, Jul 28 2004

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, multiplying rows or columns by -1 and possibly transposing.

Computed from the tables on the linked website.

N. J. A. Sloane, Tables of Hadamard matrices
Index entries for sequences related to Hadamard matrices

Cf. A007299.

a(8) from Brendan McKay, 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

A048616 Denominator of mass (Sum 1/|Aut(H)|) of Hadamard matrices of order 4n.

Original entry on oeis.org

192, 21504, 190080, 573440, 218880, 40480, 2106, 53329920

Offset: 1

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

			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. A048615, A007299, A036297.

a(8) from Brendan McKay, Feb 11 2012

A319594 Number of solutions to dft(p)^2 + dft(q)^2 = (4n-3), where p and q are even sequences of length 2n-1, p(0)=0, p(k)=+1,-1 when k<>0, q(k) is +1,-1 for all k, and dft(x) denotes the discrete Fourier transform of x.

Original entry on oeis.org

2, 4, 4, 12, 12, 0, 12, 16, 0, 36, 24, 0, 20, 36, 0, 60

Offset: 1

Jeffery Kline, Dec 16 2018

Each solution (p,q) corresponds to a family of symmetric Hadamard matrices of size 8n-4.  To construct one member from this family, set A = circulant(p) + I, B = circulant(q), C = B, D = A - 2 I and H = [ [A, B, C, D], [B, D, -A, -C], [C, -A, -D, B], [D, -C, B, -A]]. Then A, B, C and D are symmetric and H is Hadamard and symmetric.
Since p and q are assumed to be even, dft(p) and dft(q) are real-valued.
2 divides a(n) for all n. If (p,q) is a solution, then (p,-q) is also a solution.
4 divides a(n) when n>1. If (p,q) is a solution, then (+/-p,+/-q) are also solutions. When n=1, p is the length-1 sequence, (0).
It is known that a(n)>0 for n=25, 26, 29.

			For n=1, the a(1)=2 solutions are ((0),(-)) and ((0),(+)). For n=2, the a(2)=4 solutions are ((0,-,-), (-,+,+)), ((0,-,-), (+,-,-)), ((0,+,+),(-,+,+)) and ((0,+,+), (+,-,-)).

Jeffery Kline, List of all pairs (p,q) that are counted by a(n), for 1<=n<=16.
Jeffery Kline, List of pairs (p,q) that establish a(n)>0, for n=25, 26, and 29.
Jeffery Kline, Geometric Search for Hadamard Matrices, Theoret. Comput. Sci. 778 (2019), 33-46.
J. Seberry and N.A. Balonin, The Propus Construction for Symmetric Hadamard Matrices, arXiv:1512.01732 [math.CO], 2015.
J. Seberry and N.A. Balonin, Two infinite families of symmetric Hadamard matrices, Australasian Journal of Combinatorics, 69 (2015), 349-357.

Cf. A007299, A020985, A185064, A258218, A321851, A322617, A322639.

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

Original entry on oeis.org

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

N. J. A. Sloane, Oct 16 2008

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

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.

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

Cf. A007299, A036297, A016742.

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

A048885 Number of nonisomorphic orthogonal arrays OA(4n,4n-1,2,2).

Original entry on oeis.org

1, 1, 1, 5, 3, 130, 7570, 355293682

Offset: 1

N. J. A. Sloane

Equivalently, number of 3-(4*n,2*n,n-1) designs.

A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, Springer-Verlag, NY, 1999, see Table 7.37 page 165.

Index entries for sequences related to Hadamard matrices

Cf. A007299.

a(7) computed by N. J. A. Sloane

a(8) from Brendan McKay, Feb 11 2012

A321338 Number of solutions to dft(a)^2 + dft(b)^2 + dft(c)^2 + dft(d)^2 = 4n, where a,b,c,d are even +1,-1 sequences of length n and dft(x) denotes the discrete Fourier transform of x.

Original entry on oeis.org

16, 96, 64, 256, 192, 1536, 960

Offset: 1

Jeffery Kline, Dec 18 2018

Each solution corresponds to a Hadamard matrix of quaternion type. That is, if H = [[A, B, C, D], [-B, A, -D, C], [-C, D, A, -B], [-D, -C, B, A]], where A,B,C, and D are circulant matrices formed from a,b,c and d, respectively, then H is Hadamard.
Since a,b,c and d are even, their discrete Fourier transforms are real-valued.
16 is a divisor of a(n), for all n. If (a,b,c,d) is a solution, then each of the 16 tuples ((+-)a, (+-)b, (+-)c, (+-)d) is also a solution.
It appears that a(2n) > a(2n-1).
A321851(n) >= a(n), A322617(n) >= a(n) and A322639(n) >= a(n). Every solution that is counted by a(n) is also counted by A321851(n), A322617(n) and A322639(n), respectively.

L. D. Baumert and M. Hall, Hadamard matrices of the Williamson type, Math. Comp. 19:91 (1965) 442-447.
D. Z. Dokovic, Williamson matrices of order 4n for n= 33, 35, 39, Discrete mathematics (1993) May 15;115(1-3):267-71.
Jeffery Kline, A complete list of solutions (a,b,c,d), for 1<=n<=7.

Cf. A007299, A020985, A185064, A258218, A319594, A321851, A322617, A322639.

A322617 Number of solutions to |dft(a)^2 + dft(d)^2| + |dft(b)^2 + dft(c)^2| = 4n, where a,b,c,d are +1,-1 sequences of length n and dft(x) denotes the discrete Fourier transform of x.

Original entry on oeis.org

16, 96, 576, 1664, 4800, 23040

Offset: 1

Jeffery Kline, Dec 20 2018

Each solution (a,b,c,d) corresponds to a Hadamard matrix of quaternion type H = [[A, B, C, D], [-B, A, -D, C], [-C, D, A, -B], [-D, -C, B, A]], where A and D are circulant matrices formed by a and d, respectively, and B=fliplr(circulant(b)) and C=fliplr(circulant(c)). The converse is not always true. To see this, set a=(-1, -1, -1, 1), b=(-1, -1, -1, 1), c=(-1, 1, 1, 1) and d=(1, -1, -1, -1). Then H is Hadamard but |dft(a)^2 + dft(d)^2| + |dft(b)^2 + dft(c)^2| = (16, 0, 16, 0).

16 is a divisor of a(n), for all n. If (a,b,c,d) is a solution, then each of the 16 tuples ((+-)a, (+-)b, (+-)c, (+-)d) is also a solution.

L. D. Baumert and M. Hall, Hadamard matrices of the Williamson type, Math. Comp. 19:91 (1965) 442-447.
W. H. Holzmann, H. Kharaghani and B. Tayfeh-Rezaie, Williamson matrices up to order 59, Des. Codes Cryptogr. 46 (2008), 343-352.
Jeffery Kline, A complete list of solutions (a,b,c,d), for 1<=n<=5.
Jeffery Kline, Geometric Search for Hadamard Matrices, Theoret. Comput. Sci. 778 (2019), 33-46.

Cf. A007299, A020985, A185064, A258218, A319594, A321338, A321851, A322639.

A322639 Number of solutions to |dft(a)^2 + dft(b)^2 + dft(c)^2 + dft(d)^2| = 4n, where a,b,c,d are +1,-1 sequences of length n and dft(x) denotes the discrete Fourier transform of x.

Original entry on oeis.org

16, 96, 192, 896, 960, 4608, 6720

Offset: 1

Jeffery Kline, Dec 21 2018

Each solution (a,b,c,d) corresponds to a Hadamard matrix of quaternion type H = [[A, B, C, D], [-B, A, -D, C], [-C, D, A, -B], [-D, -C, B, A]], where A, B, C and D are circulant matrices formed by a, b, c and d, respectively.
16 is a divisor of a(n), for all n. If (a,b,c,d) is a solution, then each of the 16 tuples ((+-)a, (+-)b, (+-)c, (+-)d) is also a solution.
a(n) >= A321338(n). Every solution (a,b,c,d) that is counted by A321338(n) is also counted by a(n).

Jeffery Kline, A complete list of solutions (a,b,c,d), for 1<=n<=7.
Jeffery Kline, List of tuples (a,b,c,d) to demonstrate that a(n)>0, for 1<=n<=22 and n=24.

Cf. A007299, A020985, A185064, A258218, A319594, A321338, A321851, A322617.

cp's OEIS Frontend

A036297 Number of Hadamard matrices of order n.

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A096201 Number of Hadamard matrices of order 4n when a matrix and its transpose are considered equivalent.

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

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A048616 Denominator of mass (Sum 1/|Aut(H)|) of Hadamard matrices of order 4n.

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A319594 Number of solutions to dft(p)^2 + dft(q)^2 = (4n-3), where p and q are even sequences of length 2n-1, p(0)=0, p(k)=+1,-1 when k<>0, q(k) is +1,-1 for all k, and dft(x) denotes the discrete Fourier transform of x.

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

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A048885 Number of nonisomorphic orthogonal arrays OA(4n,4n-1,2,2).

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A321338 Number of solutions to dft(a)^2 + dft(b)^2 + dft(c)^2 + dft(d)^2 = 4n, where a,b,c,d are even +1,-1 sequences of length n and dft(x) denotes the discrete Fourier transform of x.

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A322617 Number of solutions to |dft(a)^2 + dft(d)^2| + |dft(b)^2 + dft(c)^2| = 4n, where a,b,c,d are +1,-1 sequences of length n and dft(x) denotes the discrete Fourier transform of x.

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A322639 Number of solutions to |dft(a)^2 + dft(b)^2 + dft(c)^2 + dft(d)^2| = 4n, where a,b,c,d are +1,-1 sequences of length n and dft(x) denotes the discrete Fourier transform of x.

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