A215723 Maximum determinant of an n X n circulant (1,-1)-matrix.
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
References
- Warren D. Smith, Posting to the Math Fun Mailing List August 18, 2012.
Links
- 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.
Crossrefs
Programs
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Maple
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: -
PARI
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 */
Extensions
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
Comments