Algorithm to get the number of iterations needed, if possible, to get an specific 2 element array
You have observed several impossible situations for $x$ and $y$, such as when $x$ and $y$ are both even or multiples of $3$, etc. More generally, we have the following characterization. For all positive integer $x$ and $y$,
$$(x,y)\text{ is reachable} \iff \gcd(x,y)=1$$
where $\gcd(x,y)$ is the greatest common divisor of $x$ and $y$. That characterization can be proved easily by mathematical induction on $n=x+y$, since $$\gcd(x,y)=\gcd(x, x+y)=\gcd(x+y,y).$$
&quot;The goal is to know how many steps are needed in order to get a given pair $(x,y)$.&quot; Here is a simple Python program that should be fast enough. It takes less than a hundredth of a second even if $x$ and $y$ have hundreds of decimal digits. The crux of the program is, count += x // y and x, y = y, x % y, which uses integer division and remainder to reduce $x$ to be smaller than $y$.
def number_of_iterations_needed(x, y):
    count = 0

    if x &lt; y:
        x, y = y, x
    # Now x &gt;= y

    while x % y != 0:
        count += x // y
        x, y = y, x % y
    # Now x % y == 0

    if y != 1:
        return None  # impossible
    else:
        count += x - y
        return count


In fact, the above program is just an augmented version of the well-known Euclidean algorithm, the granddaddy of all algorithms. Its time-complexity should be about the same as the time-complexity of the Euclidean algorithm, which is $O(\ln(\min(x,y))$. It uses about $O(1)$ space.
If you are searching for a closed form for the number of iterations needed given $x$ and $y$, you will probably not be able to find one since, as indicated on that Wikipedia page, nobody has come up with a closed form after more than two thousand years.