A003432 -id:A003432 - cp's OEIS frontend

A051752 Number of n X n (real) {0,1}-matrices having determinant A003432(n).

1, 1, 3, 3, 60, 3600, 529200, 75600, 195955200, 13716864000

Offset: 0

Eric Weisstein's World of Mathematics, Hadamard's Maximum Determinant Problem.
Eric Weisstein's World of Mathematics, (0, 1)-Matrix
Luke Zeng, Shawn Xin, Avadesian Xu, Thomas Pang, Tim Yang, Maolin Zheng, Seele's New Anti-ASIC Consensus Algorithm with Emphasis on Matrix Computation, arXiv:1905.04565 [cs.CR], 2019.
Miodrag Zivkovic, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005.
Miodrag Zivkovic, Classification of small (0,1) matrices, Linear Algebra and its Applications, 414 (2006), 310-346.

Cf. A003432, A046747, A086264, A089478, A188895.

a(5) = 3600 from Daniel P. Corson (danl(AT)MIT.EDU), Jan 09 2000
a(6) = 529200, a(7) = 75600 from Ulrich Hermisson (uhermiss(AT)rz.uni-leipzig.de), Feb 25 2003
More terms from Miodrag Zivkovic (ezivkovm(AT)matf.bg.ac.yu), Feb 28 2006
a(0)=1 prepended by Alois P. Heinz, Dec 20 2023

A089478 Triangle T(n,k) read by rows, where T(n,k) = number of times the determinant of a real n X n (0,1)-matrix takes the value k, for n >= 0, 0 <= k <= A003432(n).

Original entry on oeis.org

0, 1, 1, 1, 10, 3, 338, 84, 3, 42976, 10020, 1200, 60, 21040112, 4851360, 1213920, 144720, 43560, 3600, 39882864736, 9240051240, 3868663680, 768723480, 418703040, 63612360, 46569600, 6438600, 5014800, 529200, 292604283435872

Offset: 0

Hugo Pfoertner, Nov 04 2003

The first 4 rows were provided by Wouter Meeussen.

			a(4) = T(2,1) = 3 because there are 3 different (0,1)-matrices with determinant=1:
  ((1,0),(0,1)), ((1,1),(0,1)), ((1,0),(1,1)).
Triangle T(n,k) begins:
         0,       1;
         1,       1;
        10,       3;
       338,      84,       3;
     42976,   10020,    1200,     60;
  21040112, 4851360, 1213920, 144720, 43560, 3600;
  ...

Minfeng Wang, Table of n, a(n) for n = 0..118

Cf. T(n,0) = A046747(n), T(n,1) = A086264(n), T(n,A003432(n)) = A051752(n).
The n-th row of the table contains A089472(n) nonzero entries.
Cf. A089479.

Fortran
```
c See link in A086264.
```

Edited by Alois P. Heinz, Dec 20 2023

A094813 a(n) = number of (0,1) matrices of size n X n whose determinants are k, where -L <= k <= +L and L = A003432(n).

Original entry on oeis.org

1, 13, 10, 33, 84, 338, 84, 360, 1200, 10020, 42976, 10020, 12003600, 42795, 145485, 1206772, 4848581, 21059938, 4848585, 1206796, 145473, 42807, 3600

Offset: 1

Patricia J. Egan (capdevcom(AT)lycos.com), Jun 11 2004

			n = 2 : det([a b];[c d]) is (ad - bc) [16 possible matrices]
0 if ((a OR d) = zero) AND ((b OR c) = zero)
OR ((a AND d) = one) AND ((b AND D) = one) [10 possible matrices]
+1 if ((a AND d) = one) AND ((b OR c) = zero) [ 3 possible matrices]
-1 if ((a OR d) = zero) AND ((b AND c) = one) [ 3 possible matrices]

J. Brenner, The Hadamard maximum determinant problem, Amer. Math. Monthly, 79 (1972), 626-630.
J. Williamson, Determinants whose elements are 0 and 1, Amer. Math. Monthly 53 (1946), 427-434. Math. Rev. 8,128g.

Cf. A003432, A003433.

A007299 Number of Hadamard matrices of order 4n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

A003433 Hadamard maximal determinant problem: largest determinant of (+1,-1)-matrix of order n.

Original entry on oeis.org

1, 2, 4, 16, 48, 160, 576, 4096, 14336, 73728, 327680, 2985984, 14929920, 77635584, 418037760, 4294967296, 21474836480, 146028888064, 894426939392, 10240000000000, 59392000000000, 409600000000000

Offset: 1

N. J. A. Sloane

I added the entry for n=22 since this has been proved optimal by Chasiotis et al (reference in A003432). [Richard P. Brent, Aug 17 2021]

Ed Hughes and Rob Pratt, New Features in SAS/OR 13.1, SAS Paper SAS256-2014.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
See A003432 for further references, links and formulas.

Richard P. Brent and Judy-anne H. Osborn, On minors of maximal determinant matrices, arXiv preprint arXiv:1208.3819 [math.CO], 2012.
Thomas Decru, Tako Boris Fouotsa, Paul Frixons, Valerie Gilchrist, and Christophe Petit, Attacking trapdoors from matrix products, Cryptology ePrint Archive (2024) Paper 2024/1332.
Massimiliano Fasi and Gian Maria Negri Porzio, Determinants of Normalized Bohemian Upper Hessemberg Matrices, University of Manchester (England, 2019).
John Holbrook, Nathaniel Johnston, and Jean-Pierre Schoch, Real Schur norms and Hadamard matrices, arXiv:2206.02863 [math.CO], 2022.
Ion Nechita, Some analytical aspects of Hadamard matrices.
William P. Orrick and B. Solomon, Large-determinant sign matrices of order 4k+1, Discr. Math. 307 (2007), 226-236.
Eric Weisstein's World of Mathematics, -11-Matrix
Index entries for sequences related to binary matrices
Index entries for sequences related to Hadamard matrices
Index entries for sequences related to maximal determinants

A003432 is the main entry for this sequence.
Cf. A051753.
Cf. A188895 (number of distinct matrices having this maximal determinant).

A003432 = Cases[Import["https://oeis.org/A003432/b003432.txt", "Table"], {, }][[All, 2]];
a[n_] := 2^(n-1) A003432[[n]];
a /@ Range[21] (* Jean-François Alcover, Jan 17 2020 *)

a(n) = 2^(n-1)*A003432(n-1). E.g., a(6) = 32*A003432(5) = 32*5 = 160.

a(n) <= n^(n/2).

a(19)-a(21) added by William P. Orrick, Dec 20 2011

a(22) added by Richard P. Brent, Aug 16 2021

A089472 Number of different values taken by the determinant of a real (0,1)-matrix of order n.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 19, 43, 91, 227, 587

Offset: 0

Hugo Pfoertner, Nov 04 2003

Lower bounds: a(11) >= 1623, a(12) >= 4605, a(13) >= 14365, a(14) >= 44535, a(15) >= 145273, a(16) >= 476947

			a(7)=43 because a 7X7 (0,1)-matrix A_7 can produce the values abs(det(A_7))= {0,1,...,17,18,20,24,32}

R. Craigen, The Range of the Determinant Function on the Set of n X n (0,1)-Matrices, J. Combin. Math. Combin. Computing, 8 (1990) pp. 161-171.
W. P. Orrick, The maximal {-1, 1}-determinant of order 15.
Gerhard R. Paseman, Partial Proof of the Determinant Spectrum for 7x7 0-1 Matrices.
Miodrag Zivkovic, Massive computation as a problem solving tool, In Proceedings of the 10th Congress of Yugoslav Mathematicians (Belgrade, 2001), pages 113-128. Univ. Belgrade Fac. Math., Belgrade, 2001.
M. Zivkovic, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005.

Cf. A003432 (largest determinant of (0, 1)-matrix), A013588 (smallest integer not representable as determinant of (0, 1)-matrix), A089478 (occurrence counts), A087983 (number of different values taken by permanent of (0, 1)-matrix).

a(1)..a(4) from Wouter Meeussen.
a(7) verified by Gordon F. Royle.
Extended by William Orrick, Jan 12 2006. a(8) and a(9) computed by Miodrag Zivkovic. a(8) independently confirmed by Antonis Charalambides. a(10) computed by William Orrick.
Edited by Max Alekseyev, May 02 2011
a(0)=1 prepended by Alois P. Heinz, Mar 16 2019

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

A013588 Smallest positive integer not the determinant of an n X n {0,1}-matrix.

Original entry on oeis.org

2, 2, 3, 4, 6, 10, 19, 41, 103, 269

Offset: 1

Gerhard R. Paseman (paseman(AT)prado.com)

This majorizes the sequence of maximal determinants only up to the 6th term. It is conjectured that the sequence of maximal determinants majorizes this for all later terms.
The first term needing verification is a(11) >= 739. a(12) = 2173 has been verified by Brent, Orrick, Osborn, and Zimmermann in 2010. Lower bounds for the next terms: a(13) >= 6739, a(14) >= 21278, a(15) >= 69259, a(16) >= 230309. - Hugo Pfoertner, Jan 03 2020
Asymptotically, the sequence is at least exponential as there is an exponential lower bound of a(n) >= 2^n / (201*n) due to Shah 2022. - Rikhav Shah, Jul 09 2025

			There is no 3 X 3 {0,1}-matrix with determinant 3, as such a matrix must have a row with at least one 0 in it.

Swee Hong Chan and Igor Pak, Computational complexity of counting coincidences, arXiv:2308.10214 [math.CO], 2023. See p. 18.
R. Craigen, The Range of the Determinant Function on the Set of n X n (0,1)-Matrices, J. Combin. Math. Combin. Computing, 8 (1990) pp. 161-171.
William P. Orrick, The maximal {-1, 1}-determinant of order 15, arXiv:math/0401179 [math.CO], 2004.
William P. Orrick, Spectrum of the determinant function.
G. R. Paseman, A Different Approach to Hadamard's Maximum Determinant Problem
G. R. Paseman, Related Material
Rikhav Shah, Determinants of binary matrices achieve every integral value up to Ω(2^n/n), Linear Algebra and its Applications, Volume 645, 2022, pp. 229-236.
Miodrag Živković, Massive computation as a problem solving tool, in Proceedings of the 10th Congress of Yugoslav Mathematicians (Belgrade, 2001), pages 113-128. Univ. Belgrade Fac. Math., Belgrade, 2001.
Miodrag Živković, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005.
Index entries for sequences related to binary matrices
Index entries for sequences related to maximal determinants

Cf. A003432, A089472.

from itertools import product
from sympy import Matrix
def A013588(n):
    s, k = set(Matrix(n,n,p).det() for p in product([0,1],repeat=n**2)), 1
    while k in s:
        k += 1
    return k # Chai Wah Wu, Oct 01 2021

Extended by William P. Orrick, Jan 12 2006. a(7), a(8) and a(9) computed by Miodrag Zivkovic. a(7) and a(8) independently confirmed by Antonis Charalambides. a(10) computed by William Orrick.

A084109 n is congruent to 1 (mod 4) and is not the sum of two squares.

Original entry on oeis.org

21, 33, 57, 69, 77, 93, 105, 129, 133, 141, 161, 165, 177, 189, 201, 209, 213, 217, 237, 249, 253, 273, 285, 297, 301, 309, 321, 329, 341, 345, 357, 381, 385, 393, 413, 417, 429, 437, 453, 465, 469, 473, 489, 497

Offset: 1

William P. Orrick, Jun 18 2003

Alternatively, n is congruent to 1 (mod 4) with at least 2 distinct prime factors congruent to 3 (mod 4) in the squarefree part of n. - Comment corrected by Jean-Christophe Hervé, Oct 25 2015
Applications to the theory of optimal weighing designs and maximal determinants: An (n+1) X (n+1) conference matrix is impossible.
The upper bound of Ehlich/Wojtas on the determinant of a (0,1) matrix of order congruent to 1 (mod 4) cannot be achieved for n X n matrices.
The bound of Ehlich/Wojtas on the determinant of a (-1,1) matrix of order congruent to 2 (mod 4) cannot be achieved for (n+1) X (n+1) matrices.
Numbers with only odd prime factors, of which a strictly positive even number are raised to an odd power and congruent to 3 (mod 4). - Jean-Christophe Hervé, Oct 24 2015

			a(1) = 3*7 = 21, a(2) = 3*11 = 33, a(3) = 3*19 = 57, a(14) = 3^3*7 = 189.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 56.

Jean-Christophe Hervé, Table of n, a(n) for n = 1..1000
H. Ehlich, Determinantenabschätzungen für binäre Matrizen, Math. Z. 83 (1964) 123-132.
D. Raghavarao, Some aspects of weighing designs, Ann. Math. Stat. 31 (1960) 878-884.

Cf. A000952, A003432, A003433, A022544, A001481.

N:= 1000: # to get all entries <= N
S:= {seq(i,i=1..N,4)} minus
   {seq(seq(i^2+j^2, j=1..floor(sqrt(N-i^2)),2),i=0..floor(sqrt(N)),2)}:
sort(convert(S,list)); # Robert Israel, Oct 25 2015

a[m_] := Complement[Range[1, m, 4], Union[Flatten[Table[j^2+k^2, {j, 1, Sqrt[m], 2}, {k, 0, Sqrt[m], 2}]]]]

is(n)=if(n%4!=1, return(0)); my(f=factor(n)); for(i=1, #f~, if(f[i, 1]%4==3 && f[i, 2]%2, return(1))); 0 \\ Charles R Greathouse IV, Jul 01 2016

A119002 Maximal determinant of real n X n symmetric (0,1) matrices.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 18, 56, 144, 320

Offset: 0

Giovanni Resta, May 08 2006

The determinant of this 8 X 8 matrix is a(8) = 56:
  {0, 1, 1, 1, 0, 1, 0, 0},
  {1, 0, 1, 1, 0, 1, 0, 0},
  {1, 1, 0, 1, 0, 0, 1, 1},
  {1, 1, 1, 0, 1, 0, 0, 1},
  {0, 0, 0, 1, 1, 1, 0, 1},
  {1, 1, 0, 0, 1, 1, 1, 0},
  {0, 0, 1, 0, 0, 1, 1, 1},
  {0, 0, 1, 1, 1, 0, 1, 0}
and the determinant of this 9 X 9 matrix is a(9) = 144:
  {1, 1, 0, 1, 1, 0, 0, 1, 1},
  {1, 1, 1, 1, 0, 1, 0, 0, 0},
  {0, 1, 1, 1, 0, 0, 1, 1, 1},
  {1, 1, 1, 0, 1, 0, 1, 0, 1},
  {1, 0, 0, 1, 0, 1, 1, 0, 1},
  {0, 1, 0, 0, 1, 1, 1, 1, 0},
  {0, 0, 1, 1, 1, 1, 0, 0, 1},
  {1, 0, 1, 0, 0, 1, 0, 1, 1},
  {1, 0, 1, 1, 1, 0, 1, 1, 0}. - Jean-François Alcover, Nov 18 2017

Index entries for sequences related to maximal determinants

Cf. A003432, A119003, A119004, A118998.

a(n) <= A003432(n).

a(8) and a(9) from Jean-François Alcover, Nov 18 2017
a(0)=1 prepended by Alois P. Heinz, Nov 18 2017
a(10) from Max Alekseyev, Jun 17 2025

cp's OEIS Frontend

A051752 Number of n X n (real) {0,1}-matrices having determinant A003432(n).

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A089478 Triangle T(n,k) read by rows, where T(n,k) = number of times the determinant of a real n X n (0,1)-matrix takes the value k, for n >= 0, 0 <= k <= A003432(n).

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A094813 a(n) = number of (0,1) matrices of size n X n whose determinants are k, where -L <= k <= +L and L = A003432(n).

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A007299 Number of Hadamard matrices of order 4n.

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A003433 Hadamard maximal determinant problem: largest determinant of (+1,-1)-matrix of order n.

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A089472 Number of different values taken by the determinant of a real (0,1)-matrix of order n.

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A036297 Number of Hadamard matrices of order n.

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A013588 Smallest positive integer not the determinant of an n X n {0,1}-matrix.

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A084109 n is congruent to 1 (mod 4) and is not the sum of two squares.

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