An interesting sequence obtained from an orbits simulator: Fibonacci (II)

In the previous post I wrote an orbit simulator that produces a very intriguing sequence. I am still reviewing it, and still did not see any special property of it, but using the same orbits algorithm and instead of  using the sequence of natural numbers, using Fibonacci numbers the results are much more interesting!

Here are the results of the first terms, when the sequence is the Fibonacci numbers, due to the relationship of the Fibonacci terms, only two orbits are generated. A third orbit will never be possible!

Observing the two orbits it is easy to see that the odd terms of the first orbit divide the odd terms of the second orbit in the same position. And here is the surprise, when dividing the second orbit terms by the first orbit terms, the appearing sequence is:

{1, 7, 29, 123, 521, 2207, 9349, 39603, 167761, 710647, 3010349, 12752043, 54018521, ...}

Which is the generalized Pell equation with second term of 7, and can be found at OEIS here.

In the other hand the odd terms of the second orbit are Fibonacci(6n+5).

The second orbit appears as well as a sequence at OEIS (A015448). a(0)=1, a(1)=1, and a(n) = 4*a(n-1) + a(n-2) for n>=2.

About the first orbit, only the odd terms appear as A033887:  a(n) = Fibonacci(3n+1).

The results are quite interesting and I hope they will help me to find some relationship when using the natural numbers sequence instead of Fibonacci.