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A003433 Hadamard maximal determinant problem: largest determinant of (+1,-1)-matrix of order n.
(Formerly M1291)
12
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
Richard P. Brent and Judy-anne H. Osborn, On minors of maximal determinant matrices, arXiv preprint arXiv:1208.3819 [math.CO], 2012.
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.
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
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: A119000 A034917 A215724 * A153951 A248748 A350646
KEYWORD
nonn,hard,nice
AUTHOR
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

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Last modified March 18 22:34 EDT 2024. Contains 370951 sequences. (Running on oeis4.)