GCD of Consecutive Integers of General Fibonacci Sequence

Theorem
Let $\mathcal F = \sequence {a_n}$ be a general Fibonacci sequence generated by the parameters $r, s, t, u$:


 * $a_n = \begin{cases}

r & : n = 0 \\ s & : n = 1 \\ t a_{n - 2} + u a_{n - 1} & : n > 1 \end{cases}$

Let:
 * $d = \gcd \set {r, s}$

where $\gcd$ denotes greatest common divisor.

Let $f = \gcd \set {a_m, a_{m - 1} }$ for some $m \in \Z$.

Let $\gcd \set {f, t} = 1$.

Then:
 * $f \divides d$