A368539 -id:A368539 - cp's OEIS frontend

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}.

1, 290, 12761, 203336, 1797702, 10816592, 49651016, 186570817, 601081822

Offset: 1

Hugo Pfoertner, Jan 22 2024

a(10) >= 1714111353.

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.

			a(2) = 290: A = [1, 2;  A^3 = [37,  54;
                 3, 4]         81, 118]
.
a(3) = 12761: A = [1, 3, 4;  A^3 = [ 445,  823, 1076;
                   2, 6, 8;          890, 1646, 2152;
                   5, 7, 9]         1089, 2011, 2629]
.
a(4) = 203336: A = [1,  2,  4,  7;  A^3 = [2563,  5197,  6803,  7793;
                    3,  8,  9, 12;         5535, 11219, 14682, 16806;
                    5, 10, 13, 14;         7117, 14421, 18874, 21598;
                    6, 11, 15, 16]         8118, 16448, 21528, 24634]

Sela Fried and Toufik Mansour, On the maximal sum of the entries of a matrix power, arXiv:2308.00348 [math.CO], 2023.
Hugo Pfoertner, Examples of Solutions found by Simulated Annealing, for n = 2-10, Feb 01, 2024.

Cf. A368539.

cp's OEIS Frontend

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}.

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