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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.

%I A296106 #33 Jan 31 2018 12:34:29
%S A296106 1,3,3,8,17,8,21,130,130,21,55,931,2604,931,55,144,6871,54732,54732,
%T A296106 6871,144,377,50778
%N 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.
%C A296106 Bojagi is a puzzle game created by David Radcliffe.
%C A296106 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 A296106 Taotao Liu, Thomas Ledbetter <a href="/A296106/a296106.cs.txt">C# Program</a>
%H A296106 David Radcliffe, <a href="https://naturalmath.com/2014/10/bojagi-cute-multiplication-puzzles-by-and-for-families/">Rules of puzzle game Bojagi</a>
%F A296106 T(n,1) = A088305(n), the even-indexed Fibonacci numbers.
%F A296106 T(n,1) = Sum_{i=1..n} i*T(n-i,1) if we take T(0,1) = 1.
%e A296106 Array begins:
%e A296106 ======================================
%e A296106 n\k|   1    2     3     4    5   6
%e A296106 ---+----------------------------------
%e A296106 1  |   1    3     8    21   55 144 ...
%e A296106 2  |   3   17   130   931 6871 ...
%e A296106 3  |   8  130  2604 54732 ...
%e A296106 4  |  21  931 54732 ...
%e A296106 5  |  55 6871 ...
%e A296106 6  | 144 ...
%e A296106 ...
%e A296106 As a triangle:
%e A296106     1;
%e A296106     3,    3;
%e A296106     8,   17,     8;
%e A296106    21,  130,   130,    21;
%e A296106    55,  931,  2604,   931,   55;
%e A296106   144, 6871, 54732, 54732, 6871, 144;
%e A296106   ...
%e A296106 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 A296106 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 A296106 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 A296106 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 A296106 There are 4 boards which contain one 4. It can be placed anywhere.
%Y A296106 Cf. A088305.
%K A296106 hard,nonn,tabl,more
%O A296106 1,2
%A A296106 _Taotao Liu_, Dec 04 2017