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A372394
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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).
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1
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0, 0, 0, 33, 0, 0, 0, -77539, 1811939328, -405798912, 0, 0, 649564705105200, -2787119627540625, 86463597248512, 0, 0, 0, 353143905335474188320, -66016543975248459410178048, 0, 23092056382629010556862857216, 0, 0, 0, 0, -5310136941067623723354761986048
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OFFSET
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2,4
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COMMENTS
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Conjecture: (i) If n == 6, 8 (mod 10), and 2*n + 1 is a sum of two squares, then a(n) = 0.
(ii) If n == 5, 9 (mod 10), then a(n) is not relatively prime to 2*n + 1.
See also A372314 for other similar conjectures.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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}]];
tab={}; Do[tab=Append[tab, a[n]], {n, 2, 28}]; Print[tab]
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PROG
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(Python)
from sympy import Matrix, jacobi_symbol
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
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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