Secrets in Pascal's Triangle

Posted by Bedaywi, Mark on August 28, 2019

Read Time: 3 Minutes

When playing with the famous Pascal's Triangle one can find a lot of interesting patterns. My personal favorite has to be when you take the difference between consecutive terms and form a new triangle, A214292.

Note the triangle here takes the absolute value of the difference of consecutive terms. Looking closely one can find Catalan's Triangle hidden in the triangle. Turning the triangle 60 degrees anti-clockwise makes this revelation really pop.

When considering the terms in the triangle as the difference between consecutive terms in a sequence of binomial expansions we can derive a useful formula for each term in the sequence that even extends into the reverse triangle above the zeros.

This means we can find specific formulas for each row of the triangle and then come up with a general rule for all rows.

Individual cases of this generalisation can be found on the comments of A214292 but this provides an easy way to verify and come up with any row. This also means that formulas for catalan's triangle can be found by changing the value of n a little bit.

Another one of my favorite Pascal's Triangle properties is how it can be easily generated with a very simple game in any dimension.
The rules of the game are simple.

Ignoring the spaces in the middle Pascal's Triangle naturally emerges. We don't have to limit the checker's board to a strip. Playing on an actual board creates Pascal's Square Pyramid.

We can generalise this game to any dimension and create any variation of Pascal's Triangle. That's pretty cool.