OFFSET

1,6

COMMENTS

T(n,1) gives the number of magic squares A006052(n).

For n > 1, T(n,2*n+2) gives the number of squares with maximum heterogeneity, i.e., all sums are different (but do not necessarily form a sequence of consecutive integers), sometimes called (super)heterogeneous squares or antimagic squares.

Subsets of T(n,2) or T(n,3) with one or both of the diagonal sums not equal to the magic constant are sometimes called semimagic squares.

Sum_{k=1..2*n+2} T(n,k) = A086829(n) = (n^2)!/8 for n > 1.

REFERENCES

Pierre Berloquin, Garten der Sphinx. 150 mathematische Denkspiele, München 1984, p. 20, nr. 15 (Heterogene Quadrate), p. 20, nr. 16 (Antimagie), p. 86, nr. 148 (Höhere Antimagie), pp. 99-100, 178 (Solutions).

LINKS

Eric Weisstein's World of Mathematics, Antimagic Square.

Eric Weisstein's World of Mathematics, Magic Square.

Eric Weisstein's World of Mathematics, Semimagic Square.

EXAMPLE

T(n,k) starts with

n = 1: 1;

n = 2: 0, 0, 0, 0, 3, 0;

n = 3: 1, 22, 346, 2060, 7989, 17160, 14662, 3120;

etc.

For n = 2 there are only three square arrays up to rotation and reflection, all of heterogeneity k = 5, i.e.,

[1 2] [1 2] [1 3]

[3 4] [4 3] [4 2]

since there are always the five different sums of rows, columns and diagonals 3, 4, 5, 6 and 7.

For n = 3 the lexicographically first square arrays of heterogeneity 1 <= k <= 8 are

[2 7 6] [1 2 6] [1 2 5] [1 2 3] [1 2 3] [1 2 3] [1 2 3] [1 2 3]

[9 5 1] [5 9 4] [3 9 6] [5 6 4] [4 5 6] [4 5 7] [4 5 6] [4 5 8]

[4 3 8] [3 7 8] [4 7 8] [9 7 8] [7 8 9] [6 9 8] [7 9 8] [6 9 7]

For k = 1 we have the famous Lo Shu square with magic sum (n^3+n)/2 = 15. The other sums for the given examples are (9, 18), (8, 18, 19), (6, 15, 18, 24), (6, 12, 15, 18, 24), (6, 11, 14 16, 18, 23), (6, 12, 14, 15, 16, 17, 24) and (6, 11, 13, 14, 16, 17, 18, 22). Note that there are different sets of sums, namely a total of 6 with two values, 61 with three, 348 with four, 1295 with five, 2880 with six, 3845 with seven and 1538 with eight.

CROSSREFS

KEYWORD

nonn,more,tabf

AUTHOR

Martin Renner, Jul 27 2023

STATUS

approved