Department of Mathematics

Pascal's Triangle, Mod 2, 3, and 5.

Joint work with students David DiRico, Christopher O'Sullivan, and Stephen Tetreault

This page documents our contributions to the Online Encyclopedia of Integer Sequences, and shows how it relates to existing entries and other research.

The picture that started it all...

The spreadsheet is best viewed zoomed way out, like at 35% or so. Go to cell X1/Y1/Z1, and you can type in any modulus and the triangle will change to match. Orange cells are non-zero cells, white cells are zeros. In order to get the tringular shape, every other cell is skipped. This is just the right half of the triangle, the left half is a mirror image of the right.

I strongly encourage you to view mod 2, then mod 4, then mod 8, in succession. Also, mod 2, then mod 3, then mod 6, then back to mod 2.

Enjoy!


Mod 2

List of related sequences:

This case was quite well-documented, we add only two contributions:

Proof of first comment, in .doc format, as written by Christopher O'Sullivan

Independent Proof of first comment, in .doc format, as written by Stephen Tetreault

Chris and Steve work in opposite directions: Chris builds up from a smaller binary expansion, Steve works backwards from a larger binary expansion. Chris's approach was the one we stuck with in later attempts, below.


Mod 3 (still in progress)

List of related sequences: Changes to existing entries: Entirely new entries:

The first two are easy -- they can be generated by anyone with a little knowledge and ten minutes' patience. What is not obvious is generating a simple-form non-recursive formula that can generate the number of 1's or 2's in a given row without having to compute all of the preceding rows. That is what is contained the third and fourth entries, and what is proven in the link below.

Proof of last two new sequences, in .doc format, as written by David DiRico, and edited by M. Jaiclin (this references the Mod 2 proof above)

Mod 5 on its way


Bibliography on its way