cp's OEIS Frontend

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.

%I A359614 #23 Jan 25 2023 20:50:05
%S A359614 1,1,-3,-30,-256,-7595,-358301,-7665804,-227965955,-13089461984,
%T A359614 -2467071630448
%N 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.
%H A359614 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>.
%e A359614 a(4) = -256:
%e A359614   [   4,  3*i,  2*i,   i;
%e A359614    -3*i,    4,  3*i, 2*i;
%e A359614    -2*i, -3*i,    4, 3*i;
%e A359614      -i, -2*i, -3*i,   4 ]
%t A359614 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
%o A359614 (Python)
%o A359614 from itertools import permutations
%o A359614 from sympy import Matrix, I
%o A359614 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
%Y A359614 Cf. A350953, A359559, A359561.
%Y A359614 Cf. A359615 (maximal), A359616 (minimal permanent), A359617 (maximal permanent).
%K A359614 sign,hard,more
%O A359614 0,3
%A A359614 _Stefano Spezia_, Jan 07 2023