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

1,17

Comments

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.

Links

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

Example

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

Mathematica

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}]

Prog

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

Crossrefs

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

Sequence in context: A085355 A103120 A021433 * A124798 A318741 A171872

Adjacent sequences: A228621 A228622 A228623 * A228625 A228626 A228627

Keyword

sign

Author

Zhi-Wei Sun, Aug 28 2013

Status

approved