A086432 -id:A086432 - cp's OEIS frontend

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

W. Edwin Clark, Aug 22 2012

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

Warren D. Smith, Posting to the Math Fun Mailing List August 18, 2012.

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.

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

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

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:

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 */

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

A215724 Maximum absolute value of determinant of n X n (1,-1)-Toeplitz matrix.

Original entry on oeis.org

1, 2, 4, 16, 48, 160, 576, 2560, 12288, 73728, 327680, 2097152, 14929920, 68853760, 390905856, 2363752448

Offset: 1

W. Edwin Clark, Aug 22 2012

			a(2) = 2:
    1  1
   -1  1
a(3) = 4:
    1  1  1
   -1  1  1
    1 -1  1
a(6) = 160
    1 -1  1  1  1  1
   -1  1 -1  1  1  1
   -1 -1  1 -1  1  1
   -1 -1 -1  1 -1  1
    1 -1 -1 -1  1 -1
    1  1 -1 -1 -1  1

Warren D. Smith, Posting to the Math Fun Mailing List, August 18, 2012.

Lucas A. Brown, A215724.py.
Wikipedia, Toeplitz Matrix
Index entries for sequences related to maximal determinants

Cf. A086432 (same for circulant (0,1) matrices), A215723 (same for circulant (+1,-1) matrices).

a:=proc(n)
local T, b, U, M,d,r;
T:= combinat:-cartprod([seq({-1, 1}, j = 1..2*n-1)]);
b:= 0;
while not T[finished] do
   U := T[nextvalue]();
   M := LinearAlgebra:-ToeplitzMatrix(U,n);
   d:= abs(LinearAlgebra:-Determinant(M)):
   if d > b then b := d; end if;
end do;
return b;
end proc:

a(15) from Lucas A. Brown, Sep 06 2022

a(16) from Lucas A. Brown, Nov 03 2022

cp's OEIS Frontend

A215723 Maximum determinant of an n X n circulant (1,-1)-matrix.

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