A356490 a(n) is the determinant of a symmetric Toeplitz matrix M(n) whose first row consists of prime(1), prime(2), ..., prime(n).

1, 2, -5, 12, -19, -22, 1143, -4284, 14265, -46726, -84405, 1306096, 32312445, 522174906, 4105967871, 5135940112, -642055973735, -2832632334858, 14310549077571, 380891148658140, 4888186898996125, -49513565563840210, 383405170118692791, -2517836083641473036, -3043377347606882055

Offset: 0

Author

Stefano Spezia, Aug 09 2022

Keywords

• sign

Comments

Conjecture: abs(a(n)) is prime only for n = 1, 2, and 4.

Examples

			For n = 1 the matrix M(1) is
    2
with determinant a(1) = 2.
For n = 2 the matrix M(2) is
    2, 3
    3, 2
with determinant a(2) = -5.
For n = 3 the matrix M(3) is
    2, 3, 5
    3, 2, 3
    5, 3, 2
with determinant a(3) = 12.
		

Links

• Mathematics Stack Exchange, Determinant of a Toeplitz matrix
• Wikipedia, Toeplitz Matrix

Crossrefs

Cf. A005843 (trace of M(n)), A309131 (k-superdiagonal sum of M(n)), A356483 (hafnian of M(2*n)), A356491 (permanent of M(n)).

Cf. A350955, A350956.

Programs

Maple

A356490 := proc(n)
    local T,c ;
    if n =0 then
        return 1 ;
    end if;
    T := LinearAlgebra[ToeplitzMatrix]([seq(ithprime(c),c=1..n)],n,symmetric) ;
    LinearAlgebra[Determinant](T) ;
end proc:
seq(A356490(n),n=0..15) ; # R. J. Mathar, Jan 31 2023

Mathematica

k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Array[k, n]]; a[n_]:=Det[M[n]]; Join[{1},Table[a[n],{n,24}]]

PARI

a(n) = matdet(apply(prime, matrix(n,n,i,j,abs(i-j)+1))); \\ Michel Marcus, Aug 12 2022

Python

from sympy import Matrix, prime
def A356490(n): return Matrix(n,n,[prime(abs(i-j)+1) for i in range(n) for j in range(n)]).det() # Chai Wah Wu, Aug 12 2022

Formula

A350955(n) <= a(n) <= A350956(n).