Say that a *polynomial recurrence relation* (my terminology)
for $f\_i$ is:


* $k$ initial conditions setting $f\_1,\ldots,f\_k$ to integers ($\in \mathbb{Z})$.
* A recurrence equation of the form $f\_i =$ a polynomial in
$f\_{i-1},\ldots,f\_{i-k}$.


*Example 1*: ($k=2$): $\;f\_1=1,\; f\_2=1$, and $f\_i=f\_{i-1}+f\_{i-2}$. $\implies$ the Fibonacci sequence.
  

*Example 2*: ($k=1$): $\;f\_1=1$, and $f\_i=f\_{i-1}+1$. $\implies$ $\mathbb{N}$.



> 
> **Q1**. Is there a polynomial recurrence relation that covers $\mathbb{Z}$?
> 
> 
> 


In other words, I would like every integer (positive or negative) to be
"reached" by some $f\_i$. This may be obvious, in which case I apologize.


I arrived at this question from another direction:



> 
> **Q2**. Is there a polynomial recurrence relation over the [Gaussian
>  integers](http://en.wikipedia.org/wiki/Gaussian_integer) that covers the Gaussian integers?
> 
> 
> 



Polynomial recurrence relation covering the integers (and then Gaussian integers)
For the first question, let $f\_1=0$, $f\_2=1$, $f\_3=-1$, and $f\_i=-f\_{i-1}+f\_{i-2}+f\_{i-3}$.


