A350953 Minimal determinant of an n X n symmetric Toeplitz matrix using the integers 1 to n.

1, 1, -3, -12, -100, -1749, -47600, -800681, -39453535, -1351201968, -66984136299, -2938096403400, -235011452211680

Offset: 0

Author

Stefano Spezia, Jan 27 2022

Keywords

• sign
• hard
• more

Examples

			a(3) = -12:
    2    3    1
    3    2    3
    1    3    2
a(4) = -100:
    3    4    1    2
    4    3    4    1
    1    4    3    4
    2    1    4    3
a(5) = -1749:
    5    4    1    3    2
    4    5    4    1    3
    1    4    5    4    1
    3    1    4    5    4
    2    3    1    4    5
		

Links

• Lucas A. Brown, A350953+4+A356865.py
• Wikipedia, Toeplitz Matrix

Crossrefs

Cf. A307887, A350930, A350954 (maximal), A356865 (minimal nonzero absolute value).

Programs

Python

from itertools import permutations
from sympy import Matrix
def A350953(n): return min(Matrix([p[i:0:-1]+p[0:n-i] for i in range(n)]).det() for p in permutations(range(1,n+1))) # Chai Wah Wu, Jan 27 2022

Extensions

a(9) from Alois P. Heinz, Jan 27 2022

a(10)-a(12) from Lucas A. Brown, Sep 01 2022