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

Author

Taotao Liu, Dec 04 2017

Keywords

• hard
• nonn
• tabl
• more

Comments

Bojagi is a puzzle game created by David Radcliffe.

A Bojagi board is a rectangular board with some cells empty and some cells containing positive integers. A solution for a Bojagi board partitions the board into rectangles such that each rectangle contains exactly one integer, and that integer is the area of the rectangle.

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.