A034918 - OEIS

%I #23 Jan 26 2019 12:14:26

%S 1,1,4,16,48,128,576,2560,12288,55296,327680,2097152

%N Maximal determinant of n X n persymmetric matrix with entries {-1,0,+1}.

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

%C Conjectured: a(10) = 55296, a(11) = 327680, a(12) = 2097152. - _Jean-François Alcover_, Dec 16 2017

%H <a href="/index/De#determinants">Index entries for sequences related to maximal determinants</a>

%e For n = 1, 2, 3 use:

%e [1] [1 0] [ -1 +1 -1]

%e ... [0 1] [ +1 -1 -1]

%e ......... [ -1 -1 -1]

%t base = 3; (* base 3 is for matrix entries {-1,0,1}, base 2 is for {-1,1} *)

%t decode = Which[base == 2, 0 -> -1, base == 3, {0 -> -1, 1 -> 0, 2 -> 1}];

%t 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]];

%t 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];

%t Array[a, 9] (* _Jean-François Alcover_, Dec 16 2017 *)

%Y Cf. A034917, A034919, A034920, A034921.

%K nonn,nice,more

%O 1,3

%A _Fred Lunnon_, Dec 11 1999

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

%E Previously conjectured a(10)-a(12) confirmed by _Bert Dobbelaere_, Jan 26 2019