OFFSET
0,2
COMMENTS
Conjecture: abs(a(n)) is prime only for n = 1, 2, and 4.
LINKS
Mathematics Stack Exchange, Determinant of a Toeplitz matrix
Wikipedia, Toeplitz Matrix
EXAMPLE
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.
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}]]
PROG
(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
CROSSREFS
KEYWORD
sign
AUTHOR
Stefano Spezia, Aug 09 2022
STATUS
approved