# 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:
• A047999 Sierpinski's triangle (or gasket): triangle, read by rows, formed by reading Pascal's triangle mod 2.
• A048967 Number of even entries in row n of Pascal's triangle
• A000079 Powers of 2: a(n) = 2^n
• A001316 Gould's sequence: a(n) = Sum_{k=0..n} (C(n,k) mod 2); number of odd entries in row n of Pascal's triangle (A007318); 2^A000120(n).
• A000120 1's-counting sequence: number of 1's in binary expansion of n (or the binary weight of n).

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

• In A000079, we add the comment:
a(n) = The number of 1's in any row of Pascal's Triangle (mod 2) whose row number has exactly n 1's in its binary expansion (see A07318 and A047999) (result of putting together A001316 and A000120. [Marcus Jaiclin Jan 31 2012]
• In A047999, we add the comment:
Used to compute the total Steifel-Whitney cohomology class of the Real Projective space. This was an essential component of the proof that there are no product operations without zero divisors on R^n for n not equal to 1, 2, 4 or 8 (real numbers, complex numbers, quaternions, Cayley numbers), proven by Bott and Milnor.
and a reference connected to this where the sequence appears: Milnor, John W. and Stasheff, James D., Characteristic Classes (Princeton University Press, 1974), pp. 43-49 (sequence appears on pg. 46).

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:
• A083093 Triangle formed by reading Pascal's triangle (A007318) mod 3.
• A077267 Number of zeros in base 3 expansion of n.
• A081602 Number of 0's in ternary representation of n. [This is a duplicate of A077267, above; that is the active entry.]
• A062756 Number of 1's in ternary (base 3) expansion of n.
• A081603 Number of 2's in ternary representation of n.
• A062296 Number of entries in n-th row of Pascal's triangle divisible by 3.
• A006047 Number of entries in n-th row of Pascal's triangle not divisible by 3.
Changes to existing entries:
• To A083093, added cross-references to A062296 and A006047.
• To A077267, added cross-reference to A081603.
• To A062756, added cross-reference to A081603.
Entirely new entries:
• A206424: The number of 1's in row n of Pascal's Triangle (mod 3).
• A206425: The number of 2's in row n of Pascal's Triangle (mod 3).
• A206427: (Proposed, under review) Rectangular array, 2^(m-1)*(3^n+1) = Number of 1's in any row of Pascal's Trangle (mod 3) whose row number has exactly m 1's in its ternary expansion, and exactly n 2's in its ternary expansion (listed by anti-diagonals).
• A206428: (Proposed, under review) Rectangular array, 2^(m-1)*(3^n+1) = Number of 2's in any row of Pascal's Trangle (mod 3) whose row number has exactly m 1's in its ternary expansion, and exactly n 2's in its ternary expansion (listed by anti-diagonals).

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)