login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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

%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