A160523 a(n) is the determinant of the matrix returned by the MATLAB command magic(n).

1, -10, -360, 0, 5070000, 0, -348052801600, 0, 75035738059027200, 0, -41037749689303977660600, 0, 46138065481819513248350194800, 0, -95867954490405704140800000000000000, 0, 343347181141827635973213833586893432832000, 0, -1962493419491568854064862681564905921231239717640, 0

Offset

1,2

Comments

The MATLAB command is only supposed to be used when n >= 3. For n=2 it returns a non-magic square (none exist) with determinant -10.

[The MATLAB help page says: M = magic(n) returns an n-by-n matrix constructed from the integers 1 through n^2 with equal row and column sums. The order n must be a scalar greater than or equal to 3.]

From Robert Israel, Oct 27 2015: (Start)

For n >= 3, a(n) = 0 if and only if n is even.

a(n) is divisible by n^(n-2), because for odd n, magic(n) is the sum of an integer matrix of rank 2 and a matrix with entries divisible by n.

It appears that a(n) is divisible by n^(n-1). (End)

Links

Robert Israel, Table of n, a(n) for n = 1..210

MathWorks, MATLAB help file for magic.

Example

n=3: magic(3) = [8 1 6 / 3 5 7 / 4 9 2 ], det(magic(3)) = 8*(5*2-7*9) - 1*(3*2-4*7) + 6*(3*9-4*5) = -360.

Maple

f:= proc(n)
  if n::even then if n = 2 then return(-10) else return(0) fi fi;
  LinearAlgebra:-Determinant(Matrix(n, n, (i, j) -> n*(i + (j - (n+3)/2) mod n) + (i + (2*j-2) mod n) + 1))
end proc:
seq(f(i), i=1..30); # Robert Israel, Oct 27 2015

Mathematica

a[n_] := Which[n == 2, -10, EvenQ[n], 0, True, Det@Table[ n*Mod[i + (j - (n+3)/2), n] + Mod[i + (2j-2), n] + 1, {i, n}, {j, n}]];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, May 17 2023, after Robert Israel *)

Prog

(MATLAB) det(magic(n));

Crossrefs

Sequence in context: A016101 A249847 A185255 * A301310 A247335 A112694

Adjacent sequences: A160520 A160521 A160522 * A160524 A160525 A160526

Keyword

sign

Author

Magnus Bjerkeng (bjerkeng(AT)stud.ntnu.no), May 17 2009

Extensions

Edited by N. J. A. Sloane, May 17 2009

Further edits by Robert Israel and N. J. A. Sloane, Oct 27 2015

Status

approved