A051753 -id:A051753 - cp's OEIS frontend

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

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

A306838 Number of different values taken by the determinant of a real (-1,0,1) matrix of order n.

Original entry on oeis.org

1, 3, 5, 9, 25, 67, 233

Offset: 0

Steven E. Thornton, Mar 12 2019

Every term in this sequence is odd, since 0 is a possible determinant, and if d is a possible determinant then so is -d.

a(n) >= 1 + 2^n, since every integer determinant between -2^(n-1) and 2^(n-1) is possible (see MathOverflow link).

			For n = 2, the possible determinants of a 2x2 matrix with entries from {-1,0,1} are -2, -1, 0, 1, and 2. Since there are 5 numbers in this list, a(2) = 5.
The possible nonnegative determinants for small values of n are as follows (all the negatives of these numbers are also possible determinants):
n = 1: 0, 1
n = 2: 0, 1, 2
n = 3: 0, 1, 2, 3, 4
n = 4: 0 through 10, 12, 16
n = 5: 0 through 28, 30, 32, 36, 40, 48
n = 6: 0 through 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 120, 125, 128, 130, 132, 136, 144, 160

MathOverflow, Possible values of the determinant for matrices with elements {1,0,-1}
Steven E. Thornton, Properties of the Bohemian family of n x n matrices with population {-1, 0, 1}, Characteristic Polynomial Database.
Minfeng Wang, C++ program to calculate A306838

Number of matrices having maximum determinant is in A051753.

Edited and expanded by Nathaniel Johnston, Apr 19 2022

a(6) from Minfeng Wang, May 31 2024

cp's OEIS Frontend

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

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