TITLE: Prove every integer exists in this sequence?
QUESTION [18 upvotes]: Please could someone give me a hint on this sequences question? The question is to prove that every integer appears infinitely many times in the following sequence:
$$ \pm 1^{2} , \pm 1^{2} \pm 2^{2} , \pm 1^{2} \pm 2^{2} \pm 3^{2} , \pm 1^{2} \pm 2^{2} \pm 3^{2} \pm 4^{2} , ... $$
Any help would be much appreciated, thank you (:

REPLY [14 votes]: Hint: Note that $n^2 -(n+1)^2-(n+2)^2+(n+3)^2=4$. So by choosing signs of four consecutives appropriately, we can get $4$ or $-4$.   
Any integer is of the form $4k+r$, where $r$ is one of $0$, $1$, $2$, or $3$. Now it is just a matter of showing we can get all of $1$, $2$, and $3$. And we don't even need $3$. By the way, $2$ has a length $4$ representation.