OFFSET
1,19
COMMENTS
Conjecture: a(n) = 0 for no n > 15.
We observe that (-1)^{n*(n-1)/2}*a(n) is always a square. This is a special case of the following general result established by Zhi-Wei Sun.
Theorem: Let M = (m_{i,j}) be an n X n symmetric matrix over a commutative ring. Suppose that the (i,j)-entry m_{i,j} is zero whenever i + j is even and greater than 2. If n is even, then (-1)^{n/2}*det(M) = D(n)^2, where D(n) denotes the determinant |m_{2i,2j-1}|_{i,j = 1,...,n/2}. If n is odd, then (-1)^{(n-1)/2}*det(M) = m_{1,1}*D(n)^2, where D(n) is the determinant |m_{2i,2j+1}|_{i,j = 1,...,(n-1)/2}.
This theorem extends the result mentioned in A069191.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..200
MATHEMATICA
a[n_]:=a[n]=Det[Table[If[(i+j==2)||(Mod[i+j, 2]==1&&PrimeQ[i+j]==False), 1, 0], {i, 1, n}, {j, 1, n}]]
Table[a[n], {n, 1, 50}]
CROSSREFS
KEYWORD
sign
AUTHOR
Zhi-Wei Sun, Aug 27 2013
STATUS
approved