[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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