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

1,3

Comments

a(n) is divisible by 2^(n-1), see A215897. [Joerg Arndt, Aug 26 2012]

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.

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:

Prog

(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 */

Crossrefs

Cf. A003433, A086432 (same for circulant (0,1) matrices), A215724 (same for (1,-1)-Toeplitz matrices).

Cf. A215897 ( =a(n)/2^(n-1) ).

Sequence in context: A131126 A159964 A058922 * A034918 A119003 A220329

Adjacent sequences: A215720 A215721 A215722 * A215724 A215725 A215726

Keyword

nonn,hard

Author

W. Edwin Clark, Aug 22 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

Status

approved