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%I #53 Jun 14 2022 21:40:04
%S 1,0,4,16,48,128,512,2304,6912,22528,273408,2097152,14929920,50331648,
%T 390905856,1644167168,12279939072,69660573696,865782202368,
%U 5566277615616,41248865910784,215055782117376,2385859554836480,25783171861708800,146322302697472000,1107244165160239104,11063259546716733440,76787161889935196160
%N Maximum determinant of an n X n circulant (1,-1)-matrix.
%C a(n) is divisible by 2^(n-1), see A215897. [_Joerg Arndt_, Aug 26 2012]
%D Warren D. Smith, Posting to the Math Fun Mailing List August 18, 2012.
%H Richard P. Brent and Adam B. Yedidia, <a href="http://arxiv.org/abs/1801.00399">Computation of maximal determinants of binary circulant matrices</a>, arXiv:1801.00399 [math.CO], 2018.
%H John Holbrook, Nathaniel Johnston, and Jean-Pierre Schoch, <a href="https://arxiv.org/abs/2206.02863">Real Schur norms and Hadamard matrices</a>, arXiv:2206.02863 [math.CO], 2022.
%H N. J. A. Sloane, <a href="/A215723/a215723.txt">Table from Warren Smith's Aug 31 2012 posting to Math Fun Mailing List</a> [Gives n, a(n) and first row of matrix for n <= 28. I do not know how rigorous these results are.]
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Circulant_matrix">Circulant matrix</a>
%H <a href="/index/De#determinants">Index entries for sequences related to maximal determinants</a>
%p a:=proc(n)
%p local T, b, U, M,d,r;
%p T:= combinat:-cartprod([seq({-1, 1}, j = 1 .. n)]);
%p b:= 0;
%p while not T[finished] do
%p U := T[nextvalue]();
%p M := Matrix(n, shape = Circulant[U]);
%p d:= LinearAlgebra:-Determinant(M):
%p if d > b then b := d; end if;
%p end do;
%p return b;
%p end proc:
%o (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 */
%Y Cf. A003433, A086432 (same for circulant (0,1) matrices), A215724 (same for (1,-1)-Toeplitz matrices).
%Y Cf. A215897 ( =a(n)/2^(n-1) ).
%K nonn,hard
%O 1,3
%A _W. Edwin Clark_, Aug 22 2012
%E a(16)-a(22) from _Joerg Arndt_, Aug 25 2012
%E a(23)-a(28) (as calculated by Warren Smith) from _W. Edwin Clark_, Sep 02 2012
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