Prove every integer exists in this sequence?

sequences-and-series number-theory

Solution 1:

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.

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