A050257 - OEIS

%I #31 Dec 15 2022 14:03:00

%S 0,0,0,299710

%N Number of distinct antimagic squares of order n (modulo rotations and reflections).

%C An Anti-Magic Square (AMS) is an arrangement of the numbers 1 to n*n in a square matrix such that the row, column and diagonal sums form a sequence of consecutive integers.

%C Cormie et al. estimated that the total number of 5 X 5 antimagic squares is on the order of 10^17. However, computational evidence suggests that the number of such squares is on the order of 10^15. Out of 19 billion randomly generated 5 X 5 matrices with distinct entries in {1, 2, ..., 25}, only 6 formed antimagic squares (see Examples below). - _John M. Campbell_, Nov 27 2022

%D J. Cormie, V. Linek, S. Jiang, and R.-C. Chen, Investigating the antimagic square, J. Combin. Math. Combin. Comput., 43 (2002), 175-197.

%H J. Cormie, <a href="http://www.uwinnipeg.ca/~vlinek/jcormie/index.html">The Anti-Magic Square Project</a>

%H J. Cormie, <a href="/A050257/a050257.gif">Example</a>: sorting the sums (numbers in black on the border) yields the sequence: 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269 (from web page above).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AntimagicSquare.html">Antimagic Square.</a>

%H <a href="/index/Mag#magic">Index entries for sequences related to magic squares</a>

%e The following are antimagic squares of order 5.

%e   [22  7 20  1 18]   [ 2 22 11  6 21]

%e   [ 5  2 25 15 17]   [ 3 24 15  8 20]

%e   [13 19  6 24  3]   [12 19 14 16  7]

%e   [11  8 14 21 12]   [18  1  9 23 13]

%e   [16 23  4  9 10]   [25  5 17 10  4]

%e .

%e   [ 3 12 18 21 17]   [13 12 19  4 18]

%e   [23  5  9  4 25]   [11 22  1 21  8]

%e   [13 19 22  6  1]   [ 6 20  3 25 16]

%e   [16 14  8 24  2]   [15  5 23 10  9]

%e   [10 20 11  7 15]   [24  2 14  7 17]

%e .

%e   [ 3 24  4 20 18]   [23 22  3  1 12]

%e   [22 10 12  5 13]   [21  9 17 13  7]

%e   [17  1 21 25  6]   [ 4  5 16 24 19]

%e   [ 8  7 19 14 15]   [ 2 10 25 11 18]

%e   [ 9 23 11  2 16]   [20 14  8 15  6]

%K nonn,bref,hard,nice

%O 1,4

%A _Eric W. Weisstein_

%E a(n) not known for n >= 5.