A322150 Number of minimum shadings of an n X n Hitori solution grid as defined in A322125.

1, 4, 6, 74, 6, 900, 3230

Offset: 1

Author

Andrew Howroyd, Nov 28 2018

Keywords

• nonn
• hard
• more

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).

Examples

			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

Cf. A322125.