After a bit of experimentation, I thought of an interesting sequence of algebraic expressions defined recursively as follows:
$F_0 = 0$
$F_1 = 1$
For all integer $n \geq 2$,
$F_n = a_{n-2} F_{n-1} + F_{n-2}$.
where $a_0, a_1, a_2, a_3, \ldots$ are integer variables such that $a_0 \geq 2$ and $a_1, a_2, a_3, \ldots \geq 1$.
The first few algebraic expressions in this sequence are:
$F_0 = 0$
$F_1 = 1$
$F_2 = a_0$
$F_3 = a_1 a_0 + 1$
$F_4 = a_2 a_1 a_0 + a_2 + a_0$
$F_5 = a_3 a_2 a_1 a_0 + a_3 a_2 + a_3 a_0 + a_1 a_0 + 1$
$F_6 = a_4 a_3 a_2 a_1 a_0 + a_4 a_3 a_2 + a_4 a_3 a_0 + a_4 a_1 a_0 + a_4 + a_2 a_1 a_0 + a_2 + a_0$
$F_7 = a_5 a_4 a_3 a_2 a_1 a_0 + a_5 a_4 a_3 a_2 + a_5 a_4 a_3 a_0 + a_5 a_4 a_1 a_0 + a_5 a_4 + a_5 a_2 a_1 a_0 + a_5 a_2 + a_5 a_0 + a_3 a_2 a_1 a_0 + a_3 a_2 + a_3 a_0 + a_1 a_0 + 1$
...
I have a few questions about this sequence:
First, has this sequence of algebraic expressions been studied before? If so, what is known about this sequence? Is there a fast/direct formula or algorithm for computing $F_n$? Additionally, is this sequence useful for solving problems or for proving theorems?
Second, I conjecture that every pair $(m,n)$ of coprime nonnegative integers $m$ and $n$ can be represented uniquely as some $(F_{i-1},F_i)$ with some unique selection of valid integer values for $a_0, a_1, a_2, \ldots$ plugged in.
I also conjecture that for any $i$, the algebraic pair $(F_{i-1},F_i)$ represents the set of all pairs $(m,n)$ of coprime nonnegative integers $m$ and $n$ that require a certain number of steps $s(i)$ of the Euclidean algorithm to derive the result that $\gcd(m,n)=1$.
Are my conjectures true?
Finally, some useful resources for this sequence are below:
(1) I made a program that generates $(F_{i-1},F_i)$ for small values of $i$.
(2) Plotting $(F_{i-1},F_{i})$ pairs of coprime integers with a different color for each $i$, we get the picture below:
$(F_{i-1},F_{i})$ pairs of coprime integers" />
dark blue represents $(F_0,F_1)$,
green represents $(F_1,F_2)$,
red represents $(F_2,F_3)$,
light blue represents $(F_3,F_4)$,
purple represents $(F_4,F_5)$,
yellow represents $(F_5,F_6)$,
...
When doing the Euclidean algorithm on some pair $(m,n)$ of coprime integers $m$ and $n$, we may move from a yellow dot to a purple dot, for example, or from a purple dot to a light blue dot. But in a single step of the Euclidean algorithm, we cannot move skip a color. So, for example, we cannot move from a purple dot to a red dot directly; we must move from a purple dot to a light blue dot first, and then move from the light blue dot to a red dot. Eventually, proceeding through the colors, we will end up at the dark blue $(0,1)$ dot. Observing this picture, we may also notice other interesting patterns, such as the pattern that the ``bottom-left-most'' dot of each color is an ordered pair of consecutive Fibonacci numbers.