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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.
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%I #33 Jan 31 2018 12:34:29

%S 1,3,3,8,17,8,21,130,130,21,55,931,2604,931,55,144,6871,54732,54732,

%T 6871,144,377,50778

%N 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.

%C Bojagi is a puzzle game created by David Radcliffe.

%C 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.

%H Taotao Liu, Thomas Ledbetter <a href="/A296106/a296106.cs.txt">C# Program</a>

%H David Radcliffe, <a href="https://naturalmath.com/2014/10/bojagi-cute-multiplication-puzzles-by-and-for-families/">Rules of puzzle game Bojagi</a>

%F T(n,1) = A088305(n), the even-indexed Fibonacci numbers.

%F T(n,1) = Sum_{i=1..n} i*T(n-i,1) if we take T(0,1) = 1.

%e Array begins:

%e ======================================

%e n\k| 1 2 3 4 5 6

%e ---+----------------------------------

%e 1 | 1 3 8 21 55 144 ...

%e 2 | 3 17 130 931 6871 ...

%e 3 | 8 130 2604 54732 ...

%e 4 | 21 931 54732 ...

%e 5 | 55 6871 ...

%e 6 | 144 ...

%e ...

%e As a triangle:

%e 1;

%e 3, 3;

%e 8, 17, 8;

%e 21, 130, 130, 21;

%e 55, 931, 2604, 931, 55;

%e 144, 6871, 54732, 54732, 6871, 144;

%e ...

%e 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.

%e 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.

%e 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.

%e 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.

%e There are 4 boards which contain one 4. It can be placed anywhere.

%Y Cf. A088305.

%K hard,nonn,tabl,more

%O 1,2

%A _Taotao Liu_, Dec 04 2017