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 that covers the Gaussian integers?

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 that covers the Gaussian integers?

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: $f_1=1,\; f_2=1$, and $f_i=f_{i-1}+f_{i-2}$. $\to$ the Fibonacci sequence.
Example 2: $f_1=1$, and $f_i=f_{i-1}+1$. $\to$ $\mathbb{N}$.

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 that covers the Gaussian integers?

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 that covers the Gaussian integers?