OFFSET
0,2
COMMENTS
Conjecture: a(n) is prime only for n = 1 and 2.
Conjecture is true because a(n) is even for n >= 4. This is because all but two rows of the matrix consist of odd numbers. - Robert Israel, Oct 13 2023
LINKS
Robert Israel, Table of n, a(n) for n = 0..532
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
3, 2
2, 3
with determinant a(2) = 5.
For n = 3 the matrix M(3) is
5, 3, 2
3, 5, 3
2, 3, 5
with determinant a(3) = 51.
MAPLE
f:=proc(n) uses LinearAlgebra; local i;
Determinant(ToeplitzMatrix([seq(ithprime(i), i=n..1, -1)], symmetric));
end proc:
q(0):= 1:
map(q, [$0..25]); # Robert Israel, Oct 13 2023
MATHEMATICA
k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Reverse[Array[k, n]]]; a[n_]:=Det[M[n]]; Join[{1}, Table[a[n], {n, 21}]]
PROG
(PARI) a(n) = matdet(apply(prime, matrix(n, n, i, j, n-abs(i-j)))); \\ Michel Marcus, Aug 12 2022
CROSSREFS
KEYWORD
sign
AUTHOR
Stefano Spezia, Aug 09 2022
STATUS
approved