Fibonacci like sequences made from Pascal like triangles.

It is well known that you can produce the Fibonacci sequence from the Pascal's triangle.

Example 6:
Figure 22 is the well known Pascal's triangle. From this we can produce the Fibonacci sequence by adding numbers.

1, 1, 1+1=2, 1+2 = 3, 1+3+1=5, 1+4+3=8, 1+5+6+1=13, ...

$$\begin{array}{c} Figure(22) \\ \\ 1 \\ 1, 1 \\ 1, 2, 1\\ 1, 3... ...4, 1\\ 1, 5, 10, 10, 5, 1 \\ 1, 6, 15, 20, 15, 6, 1 \end{array}$$

The students tried the same kind of calculation for the Pascal like triangles they made, and found a very interesting fact.

Example 7:
We are going to use Figure 6 again. By using the same method we used in Example 6 we can make a sequence as the followings.

1, 1, 1+1=2, 2+2=4, 2+3+1=6, 2+5+3=10, 3+7+6+1=17,...

$$\begin{array}{c} Figure(6) \\ \\ 1 \\ 1, 1 \\ 1, 2, 1\\ 2, 3,... ...4, 1\\ 2, 7, 11, 10, 5, 1 \\ 3, 9, 18, 21, 15, 6, 1 \end{array}$$

The sequence we produced is very much like the Fibonacci sequence. The only difference is to add 1 periodically.

We can also produce Fibonacci like sequence by using other triangles made from $U (p, n, m, v)$.
The authors are now studying these Fibonacci sequence. They could make an interesting formula for these sequences. See [9].

The fact that we can produce Fibonacci like sequences are very important, because this fact shows that these triangles are really Pascal like!