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A369396 a(n) is the maximal sum of the elements of A^3 where A is a square matrix of size n whose elements are a permutation of {1, 2, ..., n^2}.

%I A369396 #10 Feb 01 2024 20:04:53
%S A369396 1,290,12761,203336,1797702,10816592,49651016,186570817,601081822
%N A369396 a(n) is the maximal sum of the elements of A^3 where A is a square matrix of size n whose elements are a permutation of {1, 2, ..., n^2}.
%C A369396 a(10) >= 1714111353.
%C A369396 The solutions up to and including n=8 meet the same criteria (a) to (d) from para 2.2 of Fried and Mansour (2023) for the square of the matrices. See the comment in A368539. However, the trace of the optimal matrices is larger than in the case of squared matrices. When n=9, the row and column totals for the same index are no longer the same, but 4 of these totals differ by 2. See linked file.
%H A369396 Sela Fried and Toufik Mansour, <a href="https://arxiv.org/abs/2308.00348">On the maximal sum of the entries of a matrix power</a>, arXiv:2308.00348 [math.CO], 2023.
%H A369396 Hugo Pfoertner, <a href="/A369396/a369396.txt">Examples of Solutions found by Simulated Annealing</a>, for n = 2-10, Feb 01, 2024.
%e A369396 a(2) = 290: A = [1, 2;  A^3 = [37,  54;
%e A369396                  3, 4]         81, 118]
%e A369396 .
%e A369396 a(3) = 12761: A = [1, 3, 4;  A^3 = [ 445,  823, 1076;
%e A369396                    2, 6, 8;          890, 1646, 2152;
%e A369396                    5, 7, 9]         1089, 2011, 2629]
%e A369396 .
%e A369396 a(4) = 203336: A = [1,  2,  4,  7;  A^3 = [2563,  5197,  6803,  7793;
%e A369396                     3,  8,  9, 12;         5535, 11219, 14682, 16806;
%e A369396                     5, 10, 13, 14;         7117, 14421, 18874, 21598;
%e A369396                     6, 11, 15, 16]         8118, 16448, 21528, 24634]
%Y A369396 Cf. A368539.
%K A369396 nonn,hard,more
%O A369396 1,2
%A A369396 _Hugo Pfoertner_, Jan 22 2024