A374242 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.

1, 1, 3, 9, 3, 1, 5, 9, 1, 1, 1, 1

Offset: 3

Author

Stefano Spezia, Jul 01 2024

Keywords

• nonn
• hard
• more

Comments

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.

Conjecture: all the terms are odd.

Examples

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

Links

• Lucas A. Brown, Python program.
• Wikipedia, Toeplitz Matrix.

Crossrefs

Cf. A356865, A374139.

Cf. A374239 (minimal), A374240 (maximal), A374241 (maximal absolute value).

Programs

Mathematica

a[n_]:=Min[Select[Table[Abs[Det[ToeplitzMatrix[Join[{1}, Part[Permutations[Range[n - 1]], i]]]]], {i, (n-1)!}],Positive]]; Array[a, 8, 3]

Extensions

a(11)-a(14) from Lucas A. Brown, Oct 10 2024