A228623 -id:A228623 - cp's OEIS frontend

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.

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, -6, 1, 8, -1, -17, 25, 13, -12, 11, 12, -11, -12, -4, 1, 1, -66, -60, -26, -13, 40, -67, -1, 82, 81, -49, -32, 68, 103, -222, 503, -39, -134

Offset: 1

Zhi-Wei Sun, Aug 28 2013

sign

Conjecture: a(n) is nonzero for any n > 21.
Zhi-Wei Sun also made the following similar conjecture:
  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.

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..400

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

SQ[n_]:=IntegerQ[Sqrt[n]]
a[n_]:=Det[Table[If[SQ[i+j]==True,1,0],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,30}]

a(n)=matdet(matrix(n,n,i,j,issquare(i+j))) \\ Ralf Stephan, Sep 17 2013

A228638 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as |i-j| is prime or not.

Original entry on oeis.org

0, 0, 0, 1, 2, -5, 12, -28, 32, -36, -44, -51, 90, -129, -572, 560, -228, -1265, -3752, -7344, -7168, -5200, -1800, 22225, -223908, -3353049, -16485870, -46205099, 57203700, 262750475, -1758254750, -4839494944, -8039222088, 8145218368, 45619374914, -1229448140891, -1178481640872, -942903100573, 1256162791268, -805110454771

Offset: 1

Zhi-Wei Sun, Aug 28 2013

sign

Conjecture: a(n) is nonzero for any n > 3.

			a(1) = 0 since |1-1| = 0 is not a prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..300

Cf. A228626, A228615, A228616, A228623, A228624.

a[n_]:=Det[Table[If[PrimeQ[Abs[i-j]]==True,1,0],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,50}]

cp's OEIS Frontend

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.

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