Ersby’s Triangle: An Update

A few weeks ago, I decided to revisit my post about Ersby’s Triangle – a semi-serious look at the multiples of square numbers needed to create Pascal’s Triangle.

I wanted to see if there were any formulae to describe the numbers along a particular diagonal. At first, I tried my original method of pen and paper and simply messing around with sequences of different powers of numbers, but nothing seemed to work. I put it to one side and forgot about it.

Then today I discovered The On-Line Encyclopedia of Integer Sequences. So I typed in a few of the diagonals (from top left to bottom right) from Ersby’s Triangle.

I soon came up with some interesting results as the first two diagonals I looked at (after the diagonal with 1, 2, 3...) both linked to the largest number of pieces you could get from a shape with each cut. The first diagonal referred to two-dimensional shapes and the next one referred to three-dimensional shapes.

So I was excited to see if that continued, and if the next diagonal referred to four-dimensional shapes.

It didn’t. But, undeterred, I kept searching.

Then I noticed that a particular paper kept being cited in the search results I was getting. I followed the link to Catalan Triangle Numbers and Binomial Coefficients by Kyu-Hwan Lee and Se-Jin Oh from the University of Connecticut. And there, tucked away on page 14, was the closest I’ve ever come to being referenced in a mathematics paper.

Amazing. If only I knew what the paper actually meant.

References:

Lee, K.H., Oh, S.J., (2016) "Catalan Triangle Numbers and Binomial Coefficients," arXiv:1601.06685v