## Partitioning numbers

October 22, 2008

A student asked me the other day the following rather homework-looking question: Given a natural number $n$, how many solutions $(x,y)$ does the equation

$n=x+2y$

have for $x$ and $y$ natural numbers?

The question has a very easy answer: Simply notice that $0\le 2y\le n$ and that any $y$ like this determines a unique $x$ such that $(x,y)$ is a solution. So, there are $\frac n2 +1$ solutions if $n$ is even (as $y$ can be any of $0,1,\dots,n/2$), and there are $\frac{n+1}2$ solutions if $n$ is odd.

I didn’t tell the student what the answer is, but I asked what he had tried so far. Among what he showed me there was a piece of paper in which somebody else had scribbled

$(1+t+t^2+t^3+\dots)(1+t^2+t^4+t^6+\dots),$

which caught my interest, and is the reason for this posting.