A296106 Square array T(n,k) n >= 1, k >= 1 read by antidiagonals: T(n, k) is the number of distinct Bojagi boards with dimensions n X k that have a unique solution.
1, 3, 3, 8, 17, 8, 21, 130, 130, 21, 55, 931, 2604, 931, 55, 144, 6871, 54732, 54732, 6871, 144, 377, 50778
Offset: 1
Examples
Array begins:
======================================
n\k| 1 2 3 4 5 6
---+----------------------------------
1 | 1 3 8 21 55 144 ...
2 | 3 17 130 931 6871 ...
3 | 8 130 2604 54732 ...
4 | 21 931 54732 ...
5 | 55 6871 ...
6 | 144 ...
...
As a triangle:
1;
3, 3;
8, 17, 8;
21, 130, 130, 21;
55, 931, 2604, 931, 55;
144, 6871, 54732, 54732, 6871, 144;
...
If n=1 or k=1, any valid board (a board whose numbers add up to the area of the board) has a unique solution.
For n=2 and k=2, there are 17 boards that have a unique solution. There is 1 board in which each of the four cells has a 1.
There are 4 boards which contain two 2's. The 2's must be adjacent (not diagonally opposite) in order for the board to have a unique solution.
There are 8 boards which contain one 2 and two 1's. The 1's must be adjacent in order for the board to have a solution. The 2 can be placed in either of the remaining two cells.
There are 4 boards which contain one 4. It can be placed anywhere.
Links
- Taotao Liu, Thomas Ledbetter C# Program
- David Radcliffe, Rules of puzzle game Bojagi
Crossrefs
Cf. A088305.
Formula
T(n,1) = A088305(n), the even-indexed Fibonacci numbers.
T(n,1) = Sum_{i=1..n} i*T(n-i,1) if we take T(0,1) = 1.
Comments