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A372394 Determinant of the matrix [Jacobi(i^2+5*i*j+5*j^2,2*n+1)]_{1 where Jacobi(a,m) denotes the Jacobi symbol (a/m).

%I A372394 #15 Apr 30 2024 17:37:46
%S A372394 0,0,0,33,0,0,0,-77539,1811939328,-405798912,0,0,649564705105200,
%T A372394 -2787119627540625,86463597248512,0,0,0,353143905335474188320,
%U A372394 -66016543975248459410178048,0,23092056382629010556862857216,0,0,0,0,-5310136941067623723354761986048
%N A372394 Determinant of the matrix [Jacobi(i^2+5*i*j+5*j^2,2*n+1)]_{1<i,j<2*n}, where Jacobi(a,m) denotes the Jacobi symbol (a/m).
%C A372394 Conjecture: (i) If n == 6, 8 (mod 10), and 2*n + 1 is a sum of two squares, then a(n) = 0.
%C A372394 (ii) If n == 5, 9 (mod 10), then a(n) is not relatively prime to 2*n + 1.
%C A372394 See also A372314 for other similar conjectures.
%H A372394 D. Krachun, F. Petrov, Z.-W. Sun, and M. Vsemirnov, <a href="https://doi.org/10.1016/j.ffa.2020.101672">On some determinants involving Jacobi symbols</a>, Finite Fields Appl. 64 (2010), Article 101672.
%H A372394 Z.-W. Sun, <a href="https://doi.org/10.1016/j.ffa.2018.12.004">On some determinants with Legendre symbol entries</a>, Finite Fields Appl. 56 (2019), 285-307.
%e A372394 a(2) = 0 since the determinant of the matrix [Jacobi(i^2+5*i*j+5*j^2,2*2+1)]_{1<i,j<2*2} = [1,1;1,1] has the value 0.
%t A372394 a[n_]:=a[n]=Det[Table[JacobiSymbol[i^2+5*i*j+5*j^2, 2n+1], {i, 2, 2n-1}, {j, 2, 2n-1}]];
%t A372394 tab={}; Do[tab=Append[tab, a[n]], {n, 2, 28}]; Print[tab]
%o A372394 (Python)
%o A372394 from sympy import Matrix, jacobi_symbol
%o A372394 def A372394(n): return Matrix(n-1<<1,n-1<<1,[jacobi_symbol(i*(i+5*j+14)+j*(5*j+30)+44,(n<<1)|1) for i in range(n-1<<1) for j in range(n-1<<1)]).det() # _Chai Wah Wu_, Apr 30 2024
%Y A372394 Cf. A000290, A001481, A372314, A372385.
%K A372394 sign
%O A372394 2,4
%A A372394 _Zhi-Wei Sun_, Apr 29 2024