A003433 - OEIS

A003433

Hadamard maximal determinant problem: largest determinant of (+1,-1)-matrix of order n.
(Formerly M1291)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

REFERENCES

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.

LINKS

Table of n, a(n) for n=1..22.

Richard P. Brent and Judy-anne H. Osborn, On minors of maximal determinant matrices, arXiv preprint arXiv:1208.3819 [math.CO], 2012-2013.

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 Hessenberg Matrices, Univ. Manchester (England, 2019).

Steven Finch, Small Matrices with Large Inverses: Unimodular 4 X 4 Cases, arXiv:2607.07688 [math.CO], 2026. See p. 2.

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, 2016.

William P. Orrick and Bruce Solomon, Large-determinant sign matrices of order 4k+1, Discr. Math. 307 (2007), 226-236.

Eric Weisstein's World of Mathematics, (-1,1)-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

FORMULA

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

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

MATHEMATICA

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 *)

CROSSREFS

A003432 is the main entry for this sequence.

Cf. A051753.

Cf. A188895 (number of distinct matrices having this maximal determinant).

Sequence in context: A034917 A119000 A215724 * A153951 A380115 A248748

Adjacent sequences: A003430 A003431 A003432 * A003434 A003435 A003436

KEYWORD

nonn,hard,nice,more

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

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

STATUS

approved