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.

Another One

• From differences of Fibonacci numbers, which are also Fibonacci numbers.

\[1, 1, 2, 3, 5, 8, 13, 21, \ldots\]

• Gives you this number triangle.

\[1\] \[1, 1\] \[1, 2, 2\] \[1, 1, 3, 3\] \[2, 2, 2, 5, 5\] \[3, 3, 3, 3, 8, 8\]

The RSD are the same Fibonacci number differences.

The RS is? \(1, 2, 5, 8, 16, 28, 51, 90, 159, 278, 484, 838, \ldots\) OEIS is pulling up nothing. Hrmm. Even if, I look at Odd values or Even values only, too.

• What if? You start with term 5 and subtract 3, gives 2. Then, 8 subtract 5, gives 3, ..
• Gives: \(2, 3, 8, 15, 30, 56, 104, \ldots\)
• This is? OEIS: A099920 • OEIS Description: \(a(n) = (n+1) \cdot F(n)\), F(n) = Fibonacci numbers.

Done.

PS. Seems similar to Les Marvin sequence: \(a(n) = F(n) + (n-1) \cdot F(n-1)\), only in the other direction.