A007299 -id:A007299 - cp's OEIS frontend

A199005 Number of inequivalent cocyclic Hadamard matrices of order 4n.

Original entry on oeis.org

1, 1, 1, 5, 3, 16, 6, 100, 35

Offset: 1

N. J. A. Sloane, Nov 01 2011

P. O'Cathain and M. Roeder, The Cocyclic Hadamard matrices of order less than 40, Des. Codes Cryptogr. 58 (1) (2011) 73-88.

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.

Cf. A007299, A199006, A199007.

A199007 Related to number of cocyclic Hadamard matrices of order 4n.

Original entry on oeis.org

192, 21504, 190080, 10838016, 16440, 790224, 64488

Offset: 1

N. J. A. Sloane, Nov 01 2011

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.

Cf. A007299, A199005, A199006.

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).

A111368 The number of maximal determinant {-1,1} matrices of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 3, 3, 3, 3, 7

Offset: 1

William P. Orrick, Nov 08 2005

The number of inequivalent maximal determinant {-1,1} matrices of order n where 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. Additional terms: a(24)=60, a(25)=78, a(28)=487. The terms a(4n) are given in sequence A007299.

R. P. Brent, W. P. Orrick, J. Osborn, and P. Zimmermann, Maximal determinants and saturated D-optimal designs of orders 19 and 37, arXiv:1112.4160 [math.CO], 2011.
J. H. E. Cohn, On the number of D-optimal designs, J. Combin. Theory Ser. A 66 (1994) 214-225.
W. P. Orrick, On the enumeration of some D-optimal designs, arXiv:math/0511141 [math.CO], 2005-2006.
W. P. Orrick and B. Solomon, The Hadamard maximal determinant problem
Warren D. Smith, Studies in Computational Geometry Motivated by Mesh Generation, Ph. D. dissertation, Princeton University (1988).
E. Spence, Ted Spence's home page, website.
J. Williamson, Determinants whose elements are 0 and 1, Amer. Math. Monthly 53 (1946), 427-434. Math. Rev. 8,128g.

Cf. A003433, A007299.

Added a(19)-a(21) and Brent et al. reference.

Edited by William P. Orrick, Dec 20 2011

A136724 Numbers divisible by 4 that are not powers of 2.

Original entry on oeis.org

12, 20, 24, 28, 36, 40, 44, 48, 52, 56, 60, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 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, 244, 248

Offset: 1

Artur Jasinski, Jan 19 2008

nonn

Primitive dimensions of Hadamard matrices which cannot be obtained as tensor powers of the primitive matrix 2 X 2 {{1,1},{1,-1}}.
5+7=12=2+4+6, 9+11=20=2+4+6+8,.. Numbers that can be expressed as sum of two or more positive consecutive odd numbers AND as sums of two or more positive consecutive even numbers. - Vladimir Joseph Stephan Orlovsky, May 11 2010
Numbers in A008586 but not in A000079. - Michel Marcus, Nov 07 2013
Nicomachus called these numbers "odd-times even". - Eric M. Schmidt, Mar 30 2019

Nicomachus, Introduction to Arithmetic, Book I, Chapter X.

Cf. A007299.

Cf. A000079, A008586.

a = {}; Do[k = 4n; If[Round[Log[k]/Log[2]] == Log[k]/Log[2],[null], AppendTo[a, k]], {n, 1, 100}]; a
Select[Range[62]*4,!IntegerQ[Log2[#]]&] (* James C. McMahon, Jun 01 2025 *)

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.

A352099 Number of inequivalent {-1,1} matrices of order n, up to permutation of rows and/or columns and multiplication of rows and/or columns by -1.

Original entry on oeis.org

1, 2, 3, 12, 39, 388, 8102, 656108, 199727714, 224693292768, 893966897828288, 12397352268917562436, 598097093939369977901540, 100707091308314174859433507948, 59497535893138933753768955970555554, 124081719421265185713331815803874814236572, 919072633264334061873768956083917736204779032768

Offset: 1

Nathaniel Johnston, May 05 2022

nonn

The equivalence operations described in the title are commonly used when discussing Hadamard matrices, for example (see A007299). See A353052 for the version of this sequence that also considers transposition as part of the equivalence relation.

Since the row and column multiplication operations can be used to force the first row and column to consist only of ones, 2^((n-1)^2) is an upper bound on this sequence. A lower bound is 2^((n-1)^2) / (n!)^2.

Eugene Nonko, Table of n, a(n) for n = 1..45
Project Euler, Problem 626: Counting Binary Matrices.

Cf. A007299, A111368, A353052.

a(8) onwards from Eugene Nonko, Nov 30 2024

A206704 Number of Hadamard matrices of order 4n that are equivalent to their transpose.

Original entry on oeis.org

1, 1, 1, 3, 3, 12, 101, 3993

Offset: 1

Brendan McKay, Feb 11 2012

Number of equivalence classes under row/column permutation and negation that contain the transpose.

Index entries for sequences related to Hadamard matrices

Cf. A007299, A036297, A096201, A206705.

a(n) = 2*A096201(n) - A007299(n).

A206705 Number of Hadamard matrices of order n that are equivalent to their transpose.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 12, 0, 0, 0, 101, 0, 0, 0, 3993, 0, 0, 0

Offset: 1

Brendan McKay, Feb 11 2012

nonn

Number of equivalence classes under row/column permutation and negation that contain the transpose.

Index entries for sequences related to Hadamard matrices

Cf. A007299, A036297, A096201, A206704.

cp's OEIS Frontend

A199005 Number of inequivalent cocyclic Hadamard matrices of order 4n.

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A199007 Related to number of cocyclic Hadamard matrices of order 4n.

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

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A111368 The number of maximal determinant {-1,1} matrices of order n.

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A136724 Numbers divisible by 4 that are not powers of 2.

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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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A352099 Number of inequivalent {-1,1} matrices of order n, up to permutation of rows and/or columns and multiplication of rows and/or columns by -1.

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A206704 Number of Hadamard matrices of order 4n that are equivalent to their transpose.

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A206705 Number of Hadamard matrices of order n that are equivalent to their transpose.

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