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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. 1
1, 3, 3, 8, 17, 8, 21, 130, 130, 21, 55, 931, 2604, 931, 55, 144, 6871, 54732, 54732, 6871, 144, 377, 50778 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..23.

Taotao Liu, Thomas Ledbetter C# Program

David Radcliffe, Rules of puzzle game Bojagi

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.

EXAMPLE

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.

CROSSREFS

Cf. A088305.

Sequence in context: A208964 A104864 A300367 * A059197 A049974 A049972

Adjacent sequences:  A296103 A296104 A296105 * A296107 A296108 A296109

KEYWORD

hard,nonn,tabl,more

AUTHOR

Taotao Liu, Dec 04 2017

STATUS

approved

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Last modified August 22 17:52 EDT 2019. Contains 326182 sequences. (Running on oeis4.)