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%I #33 May 17 2023 13:01:50
%S 1,-10,-360,0,5070000,0,-348052801600,0,75035738059027200,0,
%T -41037749689303977660600,0,46138065481819513248350194800,0,
%U -95867954490405704140800000000000000,0,343347181141827635973213833586893432832000,0,-1962493419491568854064862681564905921231239717640,0
%N a(n) is the determinant of the matrix returned by the MATLAB command magic(n).
%C 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.
%C [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.]
%C From _Robert Israel_, Oct 27 2015: (Start)
%C For n >= 3, a(n) = 0 if and only if n is even.
%C 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.
%C It appears that a(n) is divisible by n^(n-1). (End)
%H Robert Israel, <a href="/A160523/b160523.txt">Table of n, a(n) for n = 1..210</a>
%H MathWorks, <a href="http://www.mathworks.com/help/matlab/ref/magic.html">MATLAB help file for magic</a>.
%e 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.
%p f:= proc(n)
%p if n::even then if n = 2 then return(-10) else return(0) fi fi;
%p LinearAlgebra:-Determinant(Matrix(n,n,(i,j) -> n*(i + (j - (n+3)/2) mod n) + (i + (2*j-2) mod n) + 1))
%p end proc:
%p seq(f(i), i=1..30); # _Robert Israel_, Oct 27 2015
%t 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}]];
%t Table[a[n], {n, 1, 20}] (* _Jean-François Alcover_, May 17 2023, after _Robert Israel_ *)
%o (MATLAB) det(magic(n));
%K sign
%O 1,2
%A Magnus Bjerkeng (bjerkeng(AT)stud.ntnu.no), May 17 2009
%E Edited by _N. J. A. Sloane_, May 17 2009
%E Further edits by _Robert Israel_ and _N. J. A. Sloane_, Oct 27 2015
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