

A228591


Determinant of the n X n (0,1)matrix with (i,j)entry equal to 1 if and only if i + j is 2 or an odd composite number.


12



1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 9, 81, 9, 1225, 2500, 2500, 2500, 225, 121, 841, 19044, 29584, 355216, 1527696, 141376, 40000, 40000, 10000, 59536, 258064, 139876, 935089, 885481, 16384, 1876900, 1710864, 818875456, 22896531856, 23799232900, 66328911936, 158281561, 45320023225
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OFFSET

1,19


COMMENTS

Conjecture: a(n) = 0 for no n > 15.
We observe that (1)^{n*(n1)/2}*a(n) is always a square. This is a special case of the following general result established by ZhiWei 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,2j1}_{i,j = 1,...,n/2}. If n is odd, then (1)^{(n1)/2}*det(M) = m_{1,1}*D(n)^2, where D(n) is the determinant m_{2i,2j+1}_{i,j = 1,...,(n1)/2}.
This theorem extends the result mentioned in A069191.


LINKS

ZhiWei 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

Cf. A069191, A071524, A228552, A228557, A228559, A228561, A228574, A228578, A228615, A228616.
Sequence in context: A332702 A117817 A328760 * A219664 A107346 A209280
Adjacent sequences: A228588 A228589 A228590 * A228592 A228593 A228594


KEYWORD

sign


AUTHOR

ZhiWei Sun, Aug 27 2013


STATUS

approved



