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a(n) is the minimal determinant of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.
7

%I #12 Oct 10 2024 07:24:14

%S 1,1,0,-1,1,-545,-13805,-301184,-18551951,-352513176,-31451535983,

%T -1209153784888,-87868166035113,-4204963833160760,-664087819207293468

%N a(n) is the minimal determinant of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A374239%2B40%2B41%2B42.py">Python program</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>.

%e a(5) = -545:

%e [1, 1, 4, 2, 3]

%e [1, 1, 1, 4, 2]

%e [4, 1, 1, 1, 4]

%e [2, 4, 1, 1, 1]

%e [3, 2, 4, 1, 1]

%t a[0]=1; a[n_]:=Min[Table[Det[ToeplitzMatrix[Join[{1}, Part[Permutations[Range[n - 1]], i]]]], {i, (n-1)!}]]; Array[a, 11, 0]

%Y Cf. A350953, A374139.

%Y Cf. A374240 (maximal), A374241 (maximal absolute value), A374242 (minimal nonzero absolute value).

%K sign,hard,more

%O 0,6

%A _Stefano Spezia_, Jul 01 2024

%E a(11)-a(14) from _Lucas A. Brown_, Oct 10 2024