cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A372394 Determinant of the matrix [Jacobi(i^2+5*i*j+5*j^2,2*n+1)]_{1

Original entry on oeis.org

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
Offset: 2

Views

Author

Zhi-Wei Sun, Apr 29 2024

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A000290, A001481, A372314, A372385.

Programs

  • Mathematica
    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]
  • 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

A372409 Determinant of the matrix obtained from [Jacobi(i-j, 2*n+1)]_{0

Original entry on oeis.org

1, -1, 1, -3, 3, -8, -5, -5, -24, 7, 0, 0, 9, 9, -81312, -1341867, 11, -19685120, -13, -13, 0, -15, 0, -180287762432, 17, -1407939911477, 10526233598464, 19, 19, 0, 6040299856799, -21, 29830847001120768, 23, -23, 0, 115407361849089836, -25, 0, 27, 104060523591574200
Offset: 2

Views

Author

Zhi-Wei Sun, Apr 29 2024

Keywords

Comments

Conjecture 1: If 2*n + 1 (with n > 1) is a prime p, then a(n)/floor((p-2)/3) coincides with (-1)^((p+3)/4) if p == 1 (mod 4), and (-1)^((h(-p)+1)/2) if p == 3 (mod 4), where h(-p) is the class number of the imaginary quadratic field Q(sqrt(-p)).
Conjecture 2: Let p > 3 be a prime, and let S(p) and T(p) denote the matrices obtained from [Jacobi(i+j,p)]{1<=i,j<=(p-3)/2} and [Jacobi(i+j,p)]{0<=i,j<=(p-3)/2} (respectively) by replacing all the entries in the first row by 1. Then det S(p) = -det T(p) = 2^((p-5)/2)*s(p), where s(p) is (-1)^((p+3)/4) if p == 1 (mod 4), and (-1)^((h(-p)-1)/2) if p == 3 (mod 4).
Both conjectures are motivated by Conjecture 4.6 in the author's 2019 FFA paper as well as the conjectures in A372385. They have been verified for primes p < 2000.

Examples

			a(3) = -1 since the determinant of the matrix [1, 1; Jacobi(2-1,2*3+1), Jacobi(2-2,2*3+1)] = [1, 1; 1, 0] has the value -1.

References

  • L.-Y. Wang and H.-L. Wu, On certain determinants involving Legendre symbols, Ramanujan J. 58 (2022), 43-56.

Links

Crossrefs

Cf. A179071, A179072, A372385.

Programs

  • Mathematica
    a[n_]:=a[n]=Det[Table[If[i==1,1,JacobiSymbol[i-j,2*n+1]],{i,1,n-1},{j,1,n-1}]];
tab={};Do[tab=Append[tab,a[n]],{n,2,42}];Print[tab]
  • PARI
    a(n) = matdet(matrix(n-1, n-1, i, j, if (i==1, 1, kronecker(i-j, 2*n+1)))); \\ Michel Marcus, Apr 30 2024
  • Python
    from sympy import Matrix, jacobi_symbol
def A372409(n): return Matrix(n-1,n-1,[jacobi_symbol(i-j,(n<<1)|1) if i else 1 for i in range(n-1) for j in range(n-1)]).det() # Chai Wah Wu, May 01 2024
Showing 1-2 of 2 results.

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