A359614 a(n) is the minimal determinant of an n X n Hermitian Toeplitz matrix using all the integers 1, 2, ..., n and with all off-diagonal elements purely imaginary.

1, 1, -3, -30, -256, -7595, -358301, -7665804, -227965955, -13089461984, -2467071630448

Offset

0,3

Links

Table of n, a(n) for n=0..10.

Wikipedia, Toeplitz Matrix.

Example

a(4) = -256:
  [   4,  3*i,  2*i,   i;
   -3*i,    4,  3*i, 2*i;
   -2*i, -3*i,    4, 3*i;
     -i, -2*i, -3*i,   4 ]

Mathematica

a={1}; For[n=1, n<=8, n++, mn=Infinity; For[d=1, d<=n, d++, For[i=1, i<=(n-1)!, i++, If[(t=Det[ToeplitzMatrix[Join[{d}, I Part[Permutations[Drop[Range[n], {d}]], i]]]])<mn, mn=t]]]; AppendTo[a, mn]]; a

Prog

(Python)
from itertools import permutations
from sympy import Matrix, I
def A359614(n): return min(Matrix(n, n, [(d[i-j] if i>j else -d[j-i]) if i!=j else d[0]*I for i in range(n) for j in range(n)]).det()*(1, -I, -1, I)[n&3] for d in permutations(range(1, n+1))) # Chai Wah Wu, Jan 25 2023

Crossrefs

Cf. A350953, A359559, A359561.

Cf. A359615 (maximal), A359616 (minimal permanent), A359617 (maximal permanent).

Sequence in context: A043030 A178015 A120689 * A180426 A246246 A136896

Adjacent sequences: A359611 A359612 A359613 * A359615 A359616 A359617

Keyword

sign,hard,more

Author

Stefano Spezia, Jan 07 2023

Status

approved