How to write program to list all the function values without repeating? Suppose I have a (recursive) function $f:\Bbb N\rightarrow\Bbb N$ whose range is infinite, and I want to list all the function values without repeating. 

That is, if $f(0)=1,f(1)=1,f(2)=5,f(3)=6,f(4)=6,f(5)=2$, I need a program with outputs $1,5,6,2$ I need it to firstly list $1$, then $5$, then $6$ and then $2$. 

My idea is: Start from $0$, firstly output $f(0)$, and then increase $0$ by $1$ to get $1$, look at what is $f(1)$, if it is equal to $f(0)$, then we do not output it, otherwise we output $f(1)$. And then look at $f(2)$, if it is equal to at least one of $f(0),f(1)$, then we do not write it out, otherwise we output $f(2)$. But so far I found my knowledge too poor to write a program in psudocode...

My attempt:

x := 0; 

while (n &lt; x) 

{ if (f(n)=f(x)); 

 n := n+1; 

else

return f(x)} 


I think maybe this code is problematic. Could someone please tell me some way to write it? This is such a minor part of my math course that we are not taught how to write psudocodes and now I feel a bit frustrated...

Other simple ways to do it would also be appreciate. Thanks in advance to you all who may help!

To sum up, I wish to see a psudocode to out put all the values of the function:

$f':\text{n↦the nth value in the enumeration that isn't a repeat of a previous value}$
 where $f$ is a recursive function. Apologize for my poor knowlege of computing. I have no backgroud of computing and even not know what is "memory", only psudocode is readable for me... And actually I just need the psudocode program to prove that this function is recursive.

Thank a lot to you all who leave answers and comments!
You can implement your task by keeping track of the values that have appeared so far. You maintain an array $S$ that starts empty. You compute the values $f(0),f(1),f(2),\ldots$, and for each value, you check whether it is already in $S$; if so, you don't do anything; otherwise, you print it, and add it to $S$.