OFFSET
1,2
COMMENTS
Equivalently, the number of n X n binary matrices with the least possible number of 1's such that all 0's are connected and no 1 is adjacent to another and that it is not possible to set another 1 without either placing it adjacent to another 1 or disconnecting the 0's. The least possible number of 1's is given by A322125(n).
EXAMPLE
Case n=3: a(3) = 6: up to rotation and reflection there are 2 solutions:
X . . : . X .
. X . : . . .
. . . : . X .
.
Case n=5: a(5) = 6: up to rotation and reflection there are 2 solutions:
. . X . . : . . . X .
. X . X . : X . . . .
. . . . . : . . X . .
. . . . . : . . . . X
. X . X . : . X . . .
.
For an n X m grid the number of minimum shadings are as follows:
======================================================
n\m| 1 2 3 4 5 6 7 8 9 10 11 12
---+--------------------------------------------------
1 | 1 2 1 1 1 1 1 1 1 1 1 1 ...
2 | 2 4 2 12 12 4 48 32 8 160 80 16 ...
3 | 1 2 6 1 13 53 11 100 6 113 2 88 ...
4 | 1 12 1 74 11 44 139 512 1745 5764 19209 96 ...
5 | 1 12 13 11 6 3 2035 ...
6 | 1 4 53 44 3 900 90 ...
...
An interesting tight solution set occurs with the 5 X 6 grid. The 3 solutions are:
. X . . . : . . X . . : . . . X .
. . . . X : . X . X . : X . . . .
. . . X . : . . . . . : . X . . .
. X . . . : . . . . . : . . . X .
X . . . . : . X . X . : . . . . X
. . . X . : . . X . . : . X . . .
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Andrew Howroyd, Nov 28 2018
STATUS
approved