A003433 -id:A003433 - cp's OEIS frontend

A051753 Number of n X n (-1,0,1)-matrices having maximum determinant (=A003433(n)).

1, 4, 240, 384, 30720, 7372800

Offset: 1

This count includes equivalent matrices.

Minfeng Wang, C++ program to calculate A051753
Eric Weisstein's World of Mathematics, Hadamard's Maximum Determinant Problem
Eric Weisstein's World of Mathematics, (-1, 0, 1)-Matrix
Index entries for sequences related to Hadamard matrices
Index entries for sequences related to maximal determinants

a(4) from Jud McCranie, Oct 15 2000
a(5) from Giovanni Resta, Feb 20 2009
a(1)-a(5) confirmed and a(6) added by Minfeng Wang, May 29 2024

A188895 Number of n X n (real) {-1,1}-matrices having determinant A003433(n).

Original entry on oeis.org

1, 4, 96, 384, 30720, 7372800, 4335206400, 2477260800, 25684239974400, 7191587192832000

Offset: 0

Eric W. Weisstein, Apr 19 2011

Minfeng Wang, C++ program to calculate a(2) .. a(5)
Eric Weisstein's World of Mathematics, Hadamard's Maximum Determinant Problem
Eric Weisstein's World of Mathematics, (-1, 1)-Matrix

Cf. A003433, A051752.

a(n) = 2^(2n+1) * A051752(n) for n>=2. - Hugo Pfoertner and Minfeng Wang, Jan 22 2023

Offset changed to 0 by Hugo Pfoertner, Jan 23 2023

a(5)-a(9) from Minfeng Wang, Jan 22 2023

A133465 Erroneous version of A003433.

Original entry on oeis.org

1, 2, 4, 16, 44, 160, 576, 4096, 14336, 73728, 327680, 2985984, 14929920, 77635584

Offset: 1

dead

W. D. Smith, "9 Open Problems", 6th Computational Geometry Day, Nov 13 1987.

A003432 Hadamard maximal determinant problem: largest determinant of a (real) {0,1}-matrix of order n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 32, 56, 144, 320, 1458, 3645, 9477, 25515, 131072, 327680, 1114112, 3411968, 19531250, 56640625, 195312500

Offset: 0

N. J. A. Sloane

The entries are restricted to 0 and 1; the determinant is computed in the field of real numbers.
Suppose M = (m(i,j)) is an n X n matrix of real numbers. Let
a(n) = max det M subject to m(i,j) = 0 or 1 [this sequence],
g(n) = max det M subject to m(i,j) = -1 or 1 [A003433],
h(n) = max det M subject to m(i,j) = -1, 0 or 1 [A003433],
F(n) = max det M subject to 0 <= m(i,j) <= 1 [this sequence],
G(n) = max det M subject to -1 <= m(i,j) <= 1 [A003433].
Then a(n) = F(n), g(n) = h(n) = G(n), g(n) = 2^(n-1)*a(n-1). Thus all five problems are equivalent.
Hadamard proved that a(n) <= 2^(-n)*(n+1)^((n+1)/2), with equality if and only if a Hadamard matrix of order n+1 exists. Equivalently, g(n) <= n^(n/2), with equality if and only if a Hadamard matrix of order n exists. It is believed that a Hadamard matrix of order n exists if and only if n = 1, 2 or a multiple of 4 (see A036297).
We have a(21) = 195312500?, a(22) = 662671875?, and a(36) = 1200757082375992968. Furthermore, starting with a(23), many constructions are known that attain the upper bounds of Hadamard, Barba, and Ehlich-Wojtas, and are therefore maximal. See the Orrick-Solomon web site for further information. [Edited by William P. Orrick, Dec 20 2011]
The entry a(21) = 195312500 is now known to be correct. [Edited by Richard P. Brent, Aug 17 2021]

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 9*x^6 + 32*x^7 + 56*x^8 + ...
One of 2 ways to get determinant 9 with a 6 X 6 matrix, found by Williamson:
  1 0 0 1 1 0
  0 0 1 1 1 1
  1 1 1 0 0 1
  0 1 0 1 0 1
  0 1 0 0 1 1
  0 1 1 1 1 0

J. Hadamard, Résolution d'une question relative aux déterminants, Bull. des Sciences Math. 2 (1893), 240-246.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 54.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jan Brandts and A Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
J. Brenner, The Hadamard maximum determinant problem, Amer. Math. Monthly, 79 (1972), 626-630.
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. [From William P. Orrick, Dec 20 2011]
Richard P. Brent and Judy-anne H. Osborn, On minors of maximal determinant matrices, arXiv preprint arXiv:1208.3819 [math.CO], 2012.
Swee Hong Chan and Igor Pak, Computational complexity of counting coincidences, arXiv:2308.10214 [math.CO], 2023. See p. 18.
V. Chasiotis, S. Kounias, and N. Farmakis, The D-optimal saturated designs of order 22, Discrete Mathematics 341 (2018), 380-387. Corrigendum ibid 342 (2019), 2161.
H. Ehlich, Determinantenabschätzungen für binäre Matrizen, Math. Z. 83 (1964), 123-132.
H. Ehlich and K. Zeller, Binäre Matrizen, Zeit. Angew. Math. Mech., 42 (1962), 20-21.
J. Freixas and S. Kurz, On alpha-roughly weighted games, arXiv preprint arXiv:1112.2861 [math.CO], 2011.
Matthew Hudelson, Victor Klee and David Larman, Largest j-simplices in d-cubes: some relatives of the Hadamard maximum determinant problem, Proceedings of the Fourth Conference of the International Linear Algebra Society (Rotterdam, 1994). Linear Algebra Appl. 241/243 (1996), 519-598.
J. Huttenhain and C. Ikenmeyer, Binary Determinantal Complexity, arXiv:1410.8202 [cs.CC], 2014-2015.
Yongbin Li, Junwei Zi, Yan Liu and Xiaojun Zhang, A note of values of minors for Hadamard matrices, arXiv:1905.04662 [math.CO], 2019.
William P. Orrick, The maximal {-1, 1}-determinant of order 15, arXiv:math/0401179 [math.CO], 2004.
William P. Orrick and B. Solomon, Large determinant sign matrices of order 4k+1, arXiv:math/0311292 [math.CO], 2003.
William P. Orrick and B. Solomon, The Hadamard Maximal Determinant Problem (website)
William P. Orrick, B. Solomon, R. Dowdeswell and W. D. Smith, New lower bounds for the maximal determinant problem, arXiv:math/0304410 [math.CO], 2003.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Eric Weisstein's World of Mathematics, Hadamard's maximum determinant problem.
Eric Weisstein's World of Mathematics, (0, 1)-Matrix
Eric Weisstein's World of Mathematics, (-1, 0, 1)-Matrix
J. Williamson, Determinants whose elements are 0 and 1, Amer. Math. Monthly 53 (1946), 427-434. Math. Rev. 8,128g.
Luke Zeng, Shawn Xin, Avadesian Xu, Thomas Pang, Tim Yang and Maolin Zheng, Seele's New Anti-ASIC Consensus Algorithm with Emphasis on Matrix Computation, arXiv:1905.04565 [cs.CR], 2019.
Chuanming Zong, What is known about unit cubes, Bull. Amer. Math. Soc., 42 (2005), 181-211.
Index entries for sequences related to binary matrices
Index entries for sequences related to Hadamard matrices
Index entries for sequences related to maximal determinants

A003433(n) = 2^(n-1)*a(n-1). Cf. A013588, A036297, A051752.

a(18)-a(20) added by William P. Orrick, Dec 20 2011

a(21) added by Richard P. Brent, Aug 16 2021

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

A119003 Maximal determinant of real n X n symmetric (+1,-1) matrices.

Original entry on oeis.org

1, 0, 4, 16, 48, 160, 576, 4096, 14336, 65536

Offset: 1

Giovanni Resta, May 08 2006

Computation of the determinant of these two matrices:
  {-1, -1, -1, -1,  1,  1,  1, -1},
  {-1,  1, -1,  1,  1,  1, -1,  1},
  {-1, -1,  1,  1,  1, -1, -1, -1},
  {-1,  1,  1,  1, -1,  1,  1, -1},
  { 1,  1,  1, -1,  1,  1, -1, -1},
  { 1,  1, -1,  1,  1, -1,  1, -1},
  { 1, -1, -1,  1, -1,  1, -1, -1},
  {-1,  1, -1, -1, -1, -1, -1, -1}
and
  {-1,  1,  1, -1,  1, -1,  1,  1,  1},
  { 1, -1,  1, -1,  1,  1,  1,  1, -1},
  { 1,  1,  1,  1,  1, -1, -1,  1, -1},
  {-1, -1,  1,  1, -1,  1,  1, -1,  1},
  { 1,  1,  1, -1, -1, -1,  1, -1, -1},
  {-1,  1, -1,  1, -1,  1,  1,  1, -1},
  { 1,  1, -1,  1,  1,  1,  1, -1,  1},
  { 1,  1,  1, -1, -1,  1, -1,  1,  1},
  { 1, -1, -1,  1, -1, -1,  1,  1,  1}
shows that a(8) = A003433(8) = 4096 and a(9) = A003433(9) = 14336. - Jean-François Alcover, Nov 19 2017
a(n) = n^(n/2) once there exists a symmetric Hadamard matrix of order n. In particular, a(12) = 12^6, a(16) = 16^8, etc. - Max Alekseyev, Jun 17 2025

Index entries for sequences related to maximal determinants
N. A. Balonin, Y. N. Balonin, D. Z. Djokovic, D. A. Karbovskiy, M. B. Sergeev. Construction of symmetric Hadamard matrices, arXiv:1708.05098 [math.CO], 2017.
Wikipedia, Hadamard's maximal determinant problem.

Cf. A003433, A119002, A119001, A119005.

a(8) and a(9) from Jean-François Alcover, Nov 19 2017

a(10) from Max Alekseyev, Jun 17 2025

A215723 Maximum determinant of an n X n circulant (1,-1)-matrix.

Original entry on oeis.org

1, 0, 4, 16, 48, 128, 512, 2304, 6912, 22528, 273408, 2097152, 14929920, 50331648, 390905856, 1644167168, 12279939072, 69660573696, 865782202368, 5566277615616, 41248865910784, 215055782117376, 2385859554836480, 25783171861708800, 146322302697472000, 1107244165160239104, 11063259546716733440, 76787161889935196160

Offset: 1

W. Edwin Clark, Aug 22 2012

a(n) is divisible by 2^(n-1), see A215897. [Joerg Arndt, Aug 26 2012]

Warren D. Smith, Posting to the Math Fun Mailing List August 18, 2012.

Amiram Eldar, Table of n, a(n) for n = 1..52 (calculated from the b-file at A215897)
Richard P. Brent and Adam B. Yedidia, Computation of maximal determinants of binary circulant matrices, arXiv:1801.00399 [math.CO], 2018.
John Holbrook, Nathaniel Johnston, and Jean-Pierre Schoch, Real Schur norms and Hadamard matrices, arXiv:2206.02863 [math.CO], 2022.
N. J. A. Sloane, Table from Warren Smith's Aug 31 2012 posting to Math Fun Mailing List [Gives n, a(n) and first row of matrix for n <= 28. I do not know how rigorous these results are.]
Wikipedia, Circulant matrix.
Index entries for sequences related to maximal determinants.

Cf. A003433, A086432 (same for circulant (0,1) matrices), A215724 (same for (1,-1)-Toeplitz matrices).

Cf. A215897 ( =a(n)/2^(n-1) ).

a:=proc(n)
local T, b, U, M,d,r;
T:= combinat:-cartprod([seq({-1, 1}, j = 1 .. n)]);
b:= 0;
while not T[finished] do
   U := T[nextvalue]();
   M := Matrix(n, shape = Circulant[U]);
   d:= LinearAlgebra:-Determinant(M):
   if d > b then b := d; end if;
end do;
return b;
end proc:

a(n)={my(m=0); for(p=n>1, 2^(n-1)-1, m=max(m, matdet(matrix(n, n, i, j, 1-2*bittest(p, (i-j)%n))))); m} /* For illustrative purpose only: becomes slow for n>15 */ /* M. F. Hasler, Aug 25 2012 */

a(16)-a(22) from Joerg Arndt, Aug 25 2012

a(23)-a(28) (as calculated by Warren Smith) from W. Edwin Clark, Sep 02 2012

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

A215644 Full spectrum threshold for maximal determinant {+1, -1} matrices: largest order of submatrix for which the full spectrum of absolute determinant values occurs.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 4, 6, 6, 7, 6, 7, 7, 7, 8, 8, 8, 9, 8, 10

Offset: 1

Richard P. Brent and Judy-anne Osborn, Aug 18 2012

a(n) is the maximum of m(A) taken over all maximal determinant matrices A of order n, where m(A) is the maximum m such that the full spectrum of possible values (ignoring sign) occurs for the minors of order m of A.

			For n = 8 we have a(8) = 4 as a Hadamard matrix of order 8 has minors of order 4 with the full spectrum of values {0,8,16} (signs are ignored) but minors of order m > 4 do not have this property.

R. P. Brent, The Hadamard Maximal Determinant Problem
Richard P. Brent and Judy-anne H. Osborn, On minors of maximal determinant matrices, arXiv:1208.3819, 2012.
Index entries for sequences related to maximal determinants

Cf. A003432, A003433, A013588.

We calculated the first 21 terms of the sequence by an exhaustive computation of minors of known maximal determinant matrices as at August 2012.

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.

cp's OEIS Frontend

A051753 Number of n X n (-1,0,1)-matrices having maximum determinant (=A003433(n)).

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A188895 Number of n X n (real) {-1,1}-matrices having determinant A003433(n).

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A133465 Erroneous version of A003433.

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A003432 Hadamard maximal determinant problem: largest determinant of a (real) {0,1}-matrix of order n.

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

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Maple

Mathematica

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A119003 Maximal determinant of real n X n symmetric (+1,-1) matrices.

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A215723 Maximum determinant of an n X n circulant (1,-1)-matrix.

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Maple

PARI

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

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A215644 Full spectrum threshold for maximal determinant {+1, -1} matrices: largest order of submatrix for which the full spectrum of absolute determinant values occurs.

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