A034917 Minimal determinant (negated) of n X n persymmetric matrix with entries {-1,0,+1}.

1, 2, 4, 16, 48, 160, 576, 2304, 12288, 73728, 327680, 2097152

Offset

1,2

Comments

A persymmetric (or Hankel) matrix has M[ i,j ] = M[ i-k,j+k ] for all i and j (matrix is constant along antidiagonals).

Conjectures: These minimal determinant matrices have no zero entries; a(10) is 73728, a(11) = 327680 and a(12) = 2097152. - Jean-François Alcover, Dec 15 2017

Links

Table of n, a(n) for n=1..12.

Example

For n = 1, 2, 3 use the matrices:
[ -1] [ -1 +1] [ +1 -1 +1]
..... [ +1 +1] [ -1 +1 +1]
.............. [ +1 +1 +1]

Mathematica

base = 3; (* base 3 is for matrix entries {-1, 0, 1}, base 2 is for {-1, 1} *)
decode = Which[base == 2, 0 -> -1, base == 3, {0 -> -1, 1 -> 0, 2 -> 1}];
M[n_, k_] := Module[{row0, row}, row0 = PadLeft[IntegerDigits[k , base], 2 n-1] /. decode; row[i_] := RotateLeft[row0, i][[1 ;; n]]; Array[row, n]];
a[n_] := Module[{m0, d0, m, d, kmax}, {m0, d0} = {{}, Infinity}; kmax = base^(2 n - 1); Print["n = ", n, " kmax = ", kmax]; Do[m = M[n, k]; d = Det[m]; If[d < d0, Print["   k = ", k, " det = ", d]; {m0, d0} = {m, d}], {k, 0, kmax}]; Print["m0 = ", m0 // MatrixForm, " a(", n, ") = ", d0]; d0];
Array[a, 9] (* Jean-François Alcover, Dec 16 2017 *)

Crossrefs

Cf. A034918, A034919, A034920, A034921.

Sequence in context: A128903 A103435 A119000 * A215724 A003433 A153951

Adjacent sequences: A034914 A034915 A034916 * A034918 A034919 A034920

Keyword

nonn,nice,more

Author

Fred Lunnon, Dec 11 1999

Extensions

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 08 2006

Previously conjectured a(10)-a(12) confirmed by Bert Dobbelaere, Jan 26 2019

Status

approved