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A364527 Triangle read by rows giving the number of square arrays composed of the numbers from 1 to n^2, counted up to rotation and reflection, with heterogeneity k, i.e., number of k different sums of rows, columns or diagonals with 1 <= k <= 2*n+2 for n > 1.

%I A364527 #10 Aug 02 2023 14:10:14
%S A364527 1,0,0,0,0,3,0,1,22,346,2060,7989,17160,14662,3120,880
%N A364527 Triangle read by rows giving the number of square arrays composed of the numbers from 1 to n^2, counted up to rotation and reflection, with heterogeneity k, i.e., number of k different sums of rows, columns or diagonals with 1 <= k <= 2*n+2 for n > 1.
%C A364527 T(n,1) gives the number of magic squares A006052(n).
%C A364527 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.
%C A364527 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.
%C A364527 Sum_{k=1..2*n+2} T(n,k) = A086829(n) = (n^2)!/8 for n > 1.
%D A364527 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).
%H A364527 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AntimagicSquare.html">Antimagic Square</a>.
%H A364527 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MagicSquare.html">Magic Square</a>.
%H A364527 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SemimagicSquare.html">Semimagic Square</a>.
%e A364527 T(n,k) starts with
%e A364527   n = 1: 1;
%e A364527   n = 2: 0, 0, 0, 0, 3, 0;
%e A364527   n = 3: 1, 22, 346, 2060, 7989, 17160, 14662, 3120;
%e A364527 etc.
%e A364527 For n = 2 there are only three square arrays up to rotation and reflection, all of heterogeneity k = 5, i.e.,
%e A364527   [1 2] [1 2] [1 3]
%e A364527   [3 4] [4 3] [4 2]
%e A364527 since there are always the five different sums of rows, columns and diagonals 3, 4, 5, 6 and 7.
%e A364527 For n = 3 the lexicographically first square arrays of heterogeneity 1 <= k <= 8 are
%e A364527   [2 7 6] [1 2 6] [1 2 5] [1 2 3] [1 2 3] [1 2 3] [1 2 3] [1 2 3]
%e A364527   [9 5 1] [5 9 4] [3 9 6] [5 6 4] [4 5 6] [4 5 7] [4 5 6] [4 5 8]
%e A364527   [4 3 8] [3 7 8] [4 7 8] [9 7 8] [7 8 9] [6 9 8] [7 9 8] [6 9 7]
%e A364527 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.
%Y A364527 Cf. A006003, A006052, A050257, A086829, A088020.
%K A364527 nonn,more,tabf
%O A364527 1,6
%A A364527 _Martin Renner_, Jul 27 2023