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A215723 Maximum determinant of an n X n circulant (1,-1)-matrix. 3
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 (list; graph; refs; listen; history; text; internal format)
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
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
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
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

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Last modified March 1 05:26 EST 2024. Contains 370430 sequences. (Running on oeis4.)