Sierpinski like gaskets made from Pascal like triangles.

The fractions $F(p,n,m,v)$ have many interesting properties, but here we are going to study $U (p, n, m, v)$. See the following Figure 5 and 6.

We let $p = 3$ and $v = 1$, and make a triangle by $U (p, n, m, v)$ for natural numbers $n,m$ with $n \le 7$ and $m \le n$ . Then we get the following Figure 5.

$$\begin{array}{c} Figure(5)\\ U(3,1,1,1) \\ U(3,2,1,1),U(3,2,2... ...3,1),U(3,7,4,1),U(3,7,5,1),U(3,7,6,1),U(3,7,7,1) \\ \end{array}$$

If we calculate each $U (p, n, m, v)$, then we get the following Figure 6.

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

One of the authors (Matsui) studied the least nonnegative residues of numbers in Figure 6 taken modulo 2, and he got the following Figure 7. He is a freshman in a high school, and he did not know the Sierpinski Gasket. Therefore his discovery shows an amazing possibility of young students.

$$\begin{array}{c} Figure(7) \\ \\ 1 \\ 1, 1 \\ 1, 0, 1\\ 0, 1,... ..., 0, 0, 1\\ 0, 1, 1, 0, 1, 1 \\ 1, 1, 0, 1, 1, 0, 1 \end{array}$$

After he discovered Figure 7, he colored 1,0 with different colours. In the next section you can see many beautiful pictures made from these Pascal like triangles.