We were given the following A-1 problem from the 2003 Putnam Competition:
Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers, $$ n= a_1+a_2+ \cdots + a_k$$ With $k$ an arbitrary positive integer, and $a_1 \le a_2 \le \cdots \le a_k \le a_1+1$. For example, with $n=4$ there are 4 ways: 2, 2+2, 1+1+2, 1+1+1+1.
I managed to do this by induction, showing that there are always $n$ ways to partition an integer in such a way. In my combinatorics class however, we always solved integer partitioning problems with generating functions and I have been unable to construct one for this problem. I was wondering if the math.stackexchange community could help me out with this and at least give me a nudge in the right direction.