A367441 - OEIS

%I #17 Sep 18 2024 14:24:46

%S 1,1,2,4,8,19

%N 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.

%C a(8) <= 54, a(9) <= 160.

%C Apparently, a(n) is close to A365621(n+1) for n <= 8. Is this just a coincidence?

%H <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.

%e 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).

%e For n = 5, all but the square tetromino are needed to obtain all pentominoes, so a(5) = A000105(4)-1 = 4.

%e 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.

%e 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.

%Y Cf. A000105, A365621, A367435, A367440, A367443.

%K nonn,more,changed

%O 2,3

%A _Pontus von Brömssen_, Nov 18 2023