[seqfan] Help with a tree sequence

Ali Sada pemd70 at yahoo.com
Tue May 19 11:06:18 CEST 2020


Hi Everyone,

I need help, as usual, defining and finding the terms of the sequence below. 

Each positive integer n has n bonds.
1 has only one possibility which is to connect to 2. 
1/2
So, a(1) = 1.  

2 also has only one possibility. It is connected to 1 by one bond and to 3 by the second bond. 
1/2/3
So, a(2) = 1.  

After connecting to 2, 3 has two remaining bonds which gives it two possibilities: either to connect to 4 by both bonds, or to connect to 4 and 5. 

1/2/3//4

1/2/3/4
       \
        5
So, a(3) = 2. 

The first possibility of 3 keeps 4 with two remaining bonds. This gives us two possibilities:
Either both bonds are connected to 5, or one is connected to 5 and the other to 6.   

1/2/3//4//5       


1/2/3//4/5      
           \
            6
The second possibility of 3 keeps 4 with three remaining bonds. This gives us the four possibilities below:
a.    All 3 bonds are connected to 6.   


1-2-3-4///6       
        \
         5

b.    Two bonds connected to 6, and one to 7. 

             7
           /
1-2-3-4//6         
       \
       5
c.    Two bonds connected to 7 and one to 6. 

             7
           //
1-2-3-4/6
       \
        5   

d.    The three bonds are connected to 6, 7, and 8.          

       8  7
         \/
1-2-3-4/6
        \
         5                  


So, a(4) = 2+4 = 6. 

Five of 4's possibilities leave 5 with 4 remaining bonds. That gives us 5*8 = 40 possibilities. The sixth 4 possibility leaves 5 with 3 remaining bonds, and that gives us 4 possibilities.So, a(5) = 40+4 = 44

And so on. 


Best,

Ali








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