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%I #32 Jan 26 2019 12:14:43
%S 1,2,4,16,48,160,576,2304,12288,73728,327680,2097152
%N Minimal determinant (negated) 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 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
%e For n = 1, 2, 3 use the matrices:
%e [ -1] [ -1 +1] [ +1 -1 +1]
%e ..... [ +1 +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. A034918, A034919, A034920, A034921.
%K nonn,nice,more
%O 1,2
%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
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