[seqfan] The Savannah Math

[Ali Sada]

Hi Everyone,

The setup below has no real meaning in zoology. It is just a math practice. The math system is based on the following assumptions:
a)    One hungry lion (H) eats one zebra (Z) to become a fed lion (F).
b)    In one week, a fed lion becomes hungry; a hungry lion dies; and zebras don’t change.
c)    Every week, a new unknown animal (either F, H, or Z) enters the area. 

Let’s count the distinct possibilities for each week:

1.    In the first week, we have one unknown animal, so, have 3 possibilities: it could be either F, H, or Z.  So, a(1)=3

2.    In the second week, the fed lion becomes a hungry lion, and the hungry lion dies. So, we have only two possibilities left (H or Z). When another unknown animal (F or H or Z) comes, we have the following combinations: 
F+H=FH
F+Z=FZ
H+H=2H
H+Z=F
Z+Z=2Z
We have 5 distinct possibilities; so a(2)=5

3.    In the third week, the combinations of the second week change:
FH becomes H; FZ becomes F; 2H disappears; F becomes H; and 2Z remains 2Z.
So, we have only 3 distinct possibilities (F, H, and 2Z) 
When a new animal comes, the distinct possibilities become: (F, 2F, FH, FZ, F2Z, 2H, 3Z). 
So, a(3)=7. 
If we continue, we get a(4)=11, a(5)=14, a(6)=17, a(7)=22, etc.

I would really appreciate it if someone could help me find the terms of this sequence. Also, what’s the pattern here?

Best,

Ali