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A374282 a(n) is the minimal absolute value of the determinant of a nonsingular n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

%I A374282 #13 Jul 07 2024 05:47:13
%S A374282 1,4,12,2,13,16,21,4,1
%N A374282 a(n) is the minimal absolute value of the determinant of a nonsingular n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.
%C A374282 The offset is 2 because for n = 1 the matrix is null, and hence, singular.
%H A374282 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>.
%e A374282 a(5) = 2:
%e A374282   [0, 4, 1, 2, 3]
%e A374282   [4, 0, 4, 1, 2]
%e A374282   [1, 4, 0, 4, 1]
%e A374282   [2, 1, 4, 0, 4]
%e A374282   [3, 2, 1, 4, 0]
%t A374282 a[n_]:=Min[Select[Table[Abs[Det[ToeplitzMatrix[Join[{0},Part[Permutations[Range[n-1]],i]]]]],{i,(n-1)!}],Positive]]; Array[a,9,2]
%Y A374282 Cf. A085750, A358325.
%Y A374282 Cf. A085807 (minimal permanent), A374279 (minimal signed), A374280 (maximal signed), A374281 (maximal absolute value), A374283 (maximal permanent).
%K A374282 nonn,hard,more
%O A374282 2,2
%A A374282 _Stefano Spezia_, Jul 02 2024