I've noticed that when going through the steps of Euclidean GCD algorithms, very often the sum of the remainders in the steps $s_{i+1}$ and $s_{i+2}$ will be equal to the remainder in step $s_i$. However, this isn't always the case.
Example
To illustrate what I'm talking about consider this:
$$ gcd(612,93):\\ \begin{align*} s_1: 612 &= 6 \times 93 + 54 \\ s_2: 93 &= 1 \times 54 + 39 \\ s_3: 54 &= 1 \times 39 + 15 \\ s_4: 39 &= 2 \times 15 + 9 \\ s_5: 15 &= 1 \times 9 + 6 \\ s_6: 9 &= 1 \times 6 + 3 \\ s_7: 6 &= 2 \times 3 + 0 \end{align*} $$
Notice how $54=39+15$, $15=9+6$, $9=3+6$, but $39 \neq 15+9$.
Question
Is there any reason for this to happen? How to tell if the remainder will not be the sum of the remainders in two subsequent steps?
Hint: Note that this happens when you get a $1$ for a quotient. For example, it's right there in the lines $54=1\times 39+15,\, 15=1\times 9+6,\,9=1\times 6+3$. If you think of $93$ as the first remainder, then $93=1\times 54+39$.
Then the question is "why do you so often get a quotient of $1$?" This is a result of random processes. If the remainder after division: $a=bq+r$, is roughly random between $0$ and $b-1$, then the next term $b=rq_2+r_2$ will have quotient $q_2=1$ roughly 1/2 of the time - whenever $2r>b$.