Connectness

[koh]

    Hi, Seqfans
    In game of GO a kind of connectness is defined.

    1. Two stones are connected if they are next each other horizontally or vertically.            
    2. If a stone X is connected with a stone Y and Y is connected with a stone Z then X and Z are connected.
    3. Set of n stones is connected if all members of the set are connected each other.
    4. Set of m stones of same color is connected if the configuration becomes connected after k steps in game of GO.

    [Explanation]
    1.
         connected
         + o o +
         or
            +
            o
            o
            +
         not connected
         + o + +
         + + o +
         or
         + o + o +
         etc

    4.
         connected
         + o + +
         + + o +
         Because if black player tried to cut the two stones.
         + o x +
         + + o +
         White player would put a stone like this.
         + o x +
         + o o +

         not connected
         + o x +
         + x o +
         no way to connect them

         Where "o" means white stone, "x" means black stone


    I want to know number of connected configurations of stone  of same color.

    S : 1,8,124....

    n=1

         o        1

    n=2

         o o      2

         o +      2
         + o

         o +      4
         + +
         + o

    n=3

         o o o    2

         o o      4
         + o  

         o o +    8
         + + o

         o + o    4
         + o +

         o + +    2
         + o +
         + + o
         
         o o + o    4

         o o +     8
         + + +
         + + o

         o o       8  
         + +
         + o 

         + o +     4
         + + o
         o + +

         + + o +     8
         o + + o    

         o + o      4
         + + +
         + + o

         o o + +    8
         + + + o

         o o        8
         + +
         + +
         + o

         o + + o    4
         + + + +
         + + + +
         + + + o

         o + o +     8
         + + + o    

         o + + o    8
         + + + +
         + + + +
         + + o +

         o + + o    8
         + + + +
         + + + o

         o + + o    8
         + + + +
         + + o +

         + + + o    8
         + + + +
         o + + +
         + + o +

         + + + o    8
         + + + +
         + + + +
         + o o +

    Why is the following configuration of two stones not connected?

         o + o

    Because

         o x o

         + o +
         o x o

         + o +
         o x o
         + x +

         o o +
         o x o
         + x +

         o o x
         o x o
         + x +

         o o x o
         o x o +
         + x + +

         + + x +
         o o x o
         o x o +
         + x + +

         + + o +
         + + x +
         o o x o
         o x o +
         + x + +
          
    No way to connect the white stones
    



    Yasutoshi