Zipper & Seam

I have found a way to generate number triangles, which I call Zipper & Seam. • The first name is Zipper & Seam, since it resembles this in a way. Although, in a technical term, I call it the RSD1 method.

• How does it work?

Take a sequence, like the Pentagonal numbers. • 1, 5, 12, 22, 35, 51, 70, .. • Then compute their differences. (It works without doing this.) • However, for this first illustration RSD1 number triangle. I showing this simple example.

\[P_n = \frac{n (3n-1)}{2}, \textrm{for } n \geq 1\]

Which is \(3n + 1\). Let’s use those values for this first number triangle. Sequence = 4, 7, 10, 13, 16, 19, ..

This gives you the following number triangle.

\[4\] \[7, 7\] \[3, 10, 10\] \[3, 3, 13, 13\] \[3, 3, 3, 16, 16\]

Why Interesting?

Because, the Rising Shallow Diagonals are these very differences: \(4, 7, 10, 13, \ldots\)

Then, the Row Sums are different?! \(4, 14, 23, 32, 41, 50, 59, \ldots\)

• Which are? If you start with 14, 23, 32, .. Are \(9n+5\) values. OEIS: A017221

• However, these are also. OEIS Description: Numbers whose sum of digits is 5. OEIS: A052219

The values filled with 3 are the Seam. The pairs with 7, then 10, then 13, and so on, is the Zipper. • That is it.