Putnam 2003 A1

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 four ways: 4, 2+2, 1+1+2, 1+1+1+1.