Take the 2-minute tour ×  Moderator Note: This question is from
a contest which ended 1 Dec 2012.  Consider Pascal's Triangle
taken $\pmod 2$:  First few rows of Pascal's Triangle modulo 2
For simplicity, we will call a finite string of 0's and 1's
proper if it occurs in one of the rows of this modified Pascal's
triangle. (for example, 0 (row 3) and 10001 (row 5) are proper).
I've been exploring proper strings of length $n$. My professor
told me it is possible to  i) characterize explicitly all proper
strings of length $n$  and ii) Find an explicit formula for the
number of proper strings of length $n$.  But I cannot figure out
how to even begin either parts. This is a very interesting
problem, and I was wondering if someone could help me. Thank
you!  share|improve this question I would start by trying to
answer the question for small values of n. –  Jonah Sinick Nov 5
'12 at 0:54 This may be a silly question, but can you see any
strings which are not proper? –  JavaMan Nov 5 '12 at 3:34
@JavaMan The smalle