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%I #8 Jul 01 2024 19:22:16
%S 1,1,3,9,3,1,5,9
%N a(n) is the minimal absolute value of the determinant of a nonsingular n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.
%C The offset is 3 because for n = 2 the unique symmetric Toeplitz matrix having 1 on the main diagonal and 1 off-diagonal is singular.
%C Conjecture: all the terms are odd.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>.
%e a(5) = 3:
%e [1, 1, 2, 3, 4]
%e [1, 1, 1, 2, 3]
%e [2, 1, 1, 1, 2]
%e [3, 2, 1, 1, 1]
%e [4, 3, 2, 1, 1]
%t a[n_]:=Min[Select[Table[Abs[Det[ToeplitzMatrix[Join[{1}, Part[Permutations[Range[n - 1]], i]]]]], {i, (n-1)!}],Positive]]; Array[a, 8, 3]
%Y Cf. A356865, A374139.
%Y Cf. A374239 (minimal), A374240 (maximal), A374241 (maximal absolute value).
%K nonn,hard,more,new
%O 3,3
%A _Stefano Spezia_, Jul 01 2024
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