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 nonzero 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'scounting sequence: number of 1's in binary expansion of n (or the binary weight of n).
This case was quite welldocumented, 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]
and a link to this page.

In A047999, we add the comment:
Used to compute the total SteifelWhitney 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. 4349 (sequence appears on pg. 46).
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:

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 nth row of Pascal's triangle divisible by 3.

A006047
Number of entries in nth row of Pascal's triangle not divisible by 3.
Changes to existing entries:

To A083093, added crossreferences
to A062296 and A006047.

To A077267, added crossreference
to A081603.

To A062756, added crossreference
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^(m1)*(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 antidiagonals).

A206428: (Proposed, under review)
Rectangular array,
2^(m1)*(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 antidiagonals).
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 simpleform
nonrecursive 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