A367435
Let PG(n) be the graph with one node for each free n-celled polyomino and edges between nodes corresponding to polyominoes that can be obtained from each other by moving one cell, where the intermediate (the set of cells remaining when the cell to be moved is detached) is required to be a (connected) polyomino. a(n) is the number of edges in PG(n).
Original entry on oeis.org
0, 0, 1, 8, 45, 254, 1258, 6181, 28062, 125714, 550402, 2394654, 10326665
Offset: 1
A367127
Independence number of the n-omino graph defined in A098891.
Original entry on oeis.org
1, 1, 1, 2, 4, 6, 17
Offset: 1
A367441
Minimum size of a set of polyominoes with n-1 cells such that all free polyominoes with n cells can be obtained by adding one cell to one of the polyominoes in the set.
Original entry on oeis.org
1, 1, 2, 4, 8, 19
Offset: 2
For n <= 4, all polyominoes with n-1 cells are needed to obtain all polyominoes with n cells by adding one cell, so a(n) = A000105(n-1).
For n = 5, all but the square tetromino are needed to obtain all pentominoes, so a(5) = A000105(4)-1 = 4.
For n = 6, there are 5 different sets of a(6) = 8 pentominoes that are sufficient to obtain all hexominoes. One of these sets consists of the I, L, N, P, U, V, W, and Y pentominoes. The X pentomino is the only pentomino that does not appear in any of these sets. The I, L, N, and W pentominoes are needed in all such sets.
For n = 7, there are 8 different sets of a(7) = 19 hexominoes that are sufficient to obtain all heptominoes. 14 hexominoes appear in all these sets, 10 appear in none of them.
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