0,5

Otherwise said, all nonzero elements must be member of an 1 X 4 or of a 4 X 1 subarray, and these subarrays must not touch each other (along a "side", but they can touch "diagonally" as we see in n = 5, cf. Examples). - M. F. Hasler, Jul 05 2025

The original calculation was performed using BDDs. The original terms a(1)-a(13) were verified and new terms a(14)-a(16) were added as follows: This may be analyzed by realizing the problem as counting independent sets in a certain undirected graph, whose vertices are the top squares of a vertical segment, and the left hand squares of a horizontal segment, with edges connecting those that intersect or border on one another. This is then converted to a SAT problem and given to the model counter ganak. - Victor S. Miller, Jan 17 2026

M. Soos and K. Meel, Ganak, A High-Performance Versatile Model Counter

From M. F. Hasler, Jul 05 2025: (Start)

For n < 4, we can only have the zero matrix, hence a(n) = 1.

For n = 4, we have the zero matrix, or one row of '1's (=> 4 possibilities) or two rows of '1's separated by at least one zero row (=> 3 possibilities), or the same vertically, for a total of a(4) = 1 + 7 + 7 = 15 possibilities.

For n = 5, it is similar, but now each of the nonzero row- or column-submatrices of length 4 can touch the border at either side, and we can have up to 3 rows. Finally we can also have, for each of the 4 corners, the corresponding border row and column filled with 1's except for the corner itself. That yields a total of 1 + 2 * (5*2 + (3+2+1)*2^2 + 1*2^3) + 4 = 89. (End)

Maximal 6 X 6 example:

1.1.1.1.0.0

0.0.0.0.0.1

1.1.1.1.0.1

0.0.0.0.0.1

0.0.0.0.0.1

0.1.1.1.1.0

Cf. A145790, A145791, A145792, A145793, A145794, A145795, A145796.

Sequence in context: A241433 A220150 A164541 * A010822 A316224 A022707

Adjacent sequences: A145786 A145787 A145788 * A145790 A145791 A145792

nonn,more

R. H. Hardin, Oct 19 2008

a(14)-a(16) from Victor S. Miller, Jan 16 2026

approved