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A228624 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j is a square or not. 2

%I #15 Nov 14 2013 03:30:43

%S 0,0,-1,0,1,0,0,1,1,1,0,-1,1,0,0,-1,2,3,-3,-1,0,1,-1,-2,-5,13,-7,-7,

%T -6,1,8,-1,-17,25,13,-12,11,12,-11,-12,-4,1,1,-66,-60,-26,-13,40,-67,

%U -1,82,81,-49,-32,68,103,-222,503,-39,-134

%N Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j is a square or not.

%C Conjecture: a(n) is nonzero for any n > 21.

%C Zhi-Wei Sun also made the following similar conjecture:

%C Let A(n) be the n X n determinant with (i,j)-entry equal to 1 or 0 according as i + j is a cube or not. Then A(n) is nonzero for any n > 176.

%H Zhi-Wei Sun, <a href="/A228624/b228624.txt">Table of n, a(n) for n = 1..400</a>

%e a(1) = 0 since 1 + 1 = 2 is not a square.

%t SQ[n_]:=IntegerQ[Sqrt[n]]

%t a[n_]:=Det[Table[If[SQ[i+j]==True,1,0],{i,1,n},{j,1,n}]]

%t Table[a[n],{n,1,30}]

%o (PARI) a(n)=matdet(matrix(n,n,i,j,issquare(i+j))) \\ _Ralf Stephan_, Sep 17 2013

%Y Cf. A000290, A069191, A228591, A228557, A228559, A228615, A228616, A228623.

%K sign

%O 1,17

%A _Zhi-Wei Sun_, Aug 28 2013

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Last modified March 28 12:26 EDT 2024. Contains 371254 sequences. (Running on oeis4.)