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%I #22 Aug 28 2013 03:00:38
%S 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,
%T -225,-121,841,19044,-29584,-355216,1527696,141376,-40000,-40000,
%U 10000,59536,-258064,-139876,935089,885481,-16384,-1876900,1710864,818875456,-22896531856,-23799232900,66328911936,158281561,-45320023225
%N 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.
%C Conjecture: a(n) = 0 for no n > 15.
%C 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.
%C 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}.
%C This theorem extends the result mentioned in A069191.
%H Zhi-Wei Sun, <a href="/A228591/b228591.txt">Table of n, a(n) for n = 1..200</a>
%t 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}]]
%t Table[a[n],{n,1,50}]
%Y Cf. A069191, A071524, A228552, A228557, A228559, A228561, A228574, A228578, A228615, A228616.
%K sign
%O 1,19
%A _Zhi-Wei Sun_, Aug 27 2013
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