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

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• nice

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

• 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

Crossrefs

A003432 is the main entry for this sequence.

Cf. A051753.

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

Programs

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 *)

Formula

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

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

Extensions

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

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