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.

1, 4, 12, 2, 13, 16, 21, 4, 1

Offset: 2

Author

Stefano Spezia, Jul 02 2024

Keywords

• nonn
• hard
• more

Comments

The offset is 2 because for n = 1 the matrix is null, and hence, singular.

Examples

			a(5) = 2:
  [0, 4, 1, 2, 3]
  [4, 0, 4, 1, 2]
  [1, 4, 0, 4, 1]
  [2, 1, 4, 0, 4]
  [3, 2, 1, 4, 0]
		

Links

• Wikipedia, Toeplitz Matrix.

Crossrefs

Cf. A085750, A358325.

Cf. A085807 (minimal permanent), A374279 (minimal signed), A374280 (maximal signed), A374281 (maximal absolute value), A374283 (maximal permanent).

Programs

Mathematica

a[n_]:=Min[Select[Table[Abs[Det[ToeplitzMatrix[Join[{0},Part[Permutations[Range[n-1]],i]]]]],{i,(n-1)!}],Positive]]; Array[a,9,2]