A131519

[koh]

    Hi,Max

    You wrote : 

>>>                  Figure of G_5 : o=o=o=o=o
>This sequence is A131519. Please double check the value A131519(3)=66.Could you list all partitions?I've got a different value A131519(3)=46 and that this sequencesatisfies the recurrent formula:A131519(n) = 9*A131519(n-1) - 10*A131519(n-2) + 8*A131519(n-3)giving the following values:1, 6, 46, 362,
 2846, 22362, ...


> I propose a new formula:
>
> A131519(n) = 11*A131519(n-1) - 24*A131519(n-3) for n>3.

>Ops. It should be n>4. This formula does not work for n=4 (and it is
>not clear how u(1), v(1), w(1), and z(1) from previous email should be
>defined).

> And the sequence is
>
> 1, 6, 66, 714, 7710, ...



    My early result a(3)=66 and yours 46 seem to be incorrect.        

    I list all parrtitions.
    I classified them with length of the paths.

         G_3

         [Names of nodes and  edges]

         x   y   z     t_1=xy , t_2=yz
         o===o===o     b_1=xy , b_2=yz


         [Example of figure]

         path b_1,t_2 is represented as follows

         x   y-->z
         .-->.   .

         cycle path t_1,-b_2

         x-->.   .
         .<--.   .


    o {4}                                   {n} means directed path of length n
         x-->.-->.
         .<--.<--.    
         {t or b}*{t or b}*{x or z}         last term is for starting point
         =2^3=8

         .-->y-->.
         .<--.<--.
         {r or l}*{t or b}*{t or b}         r or l means number of symmetry
         =2^3=8

    o {3} + {1}
         x-->.-->.   x   .   . 
         .   .<--. + .-->.   .
         the same as {4}
         =8+8

    o {2} + {2}
         x-->.   .   .   y-->.
         .<--.   . + .   .<--.
         {t or b}*{t or b}*{x or z}
         =2^3=8

         x-->.   .   .   .<--z
         .<--.   . + .   .-->.
         {t or b}*{t or b}
         =2^2=4

         x-->.-->.   .   .   .
         .   .   . + .-->.-->.
         {t or b}*{t or b}*{x or z}
         =2^3=8


    o {2} + {1} + {1}
         x-->.   .   .   y-->.    .   .   .
         .<--.   . + .   .   . +  .   .-->.
         {t or b}*{y or z}*{r or l}
         =2^3=8

    o {1} + {1} + {1} + {1}
         .-->.   .   .   .<--.   .   .   .   .   .   .
         .   .   . + .   .   . + .-->.   . + .   .<--.             
         =2

    So,               
         a(3) = 8 + 8 + 8 + 8 + 8 + 4 + 8 + 8 + 2
              = 62
   
    Could you explane how to get your formula?

    Yasutoshi