Given a sequence of numbers, https://oeis.org/A001906
[0, 1, 3, 8, 21, 55, 144, 377]
What would be the proper mathematical way to define this number set, and how would it be resolved explicitly from scratch?
The keen eyed among you may have already noticed that this is every other sequence from the Fibonacci sequence. As such there is a recursive part (The Fibonacci sequence), http://oeis.org/A000045
F(0) = 0
F(1) = 1
F(n) = F(n-1) + F(n-2)
more over there is a sub pattern in this case ever 2nd number of the sequence. http://oeis.org/A005843
Xn = 2n
How does one derive the recursive formula for F2n, where Fn is the nth Fibonacci number as F(n) = F(2n-1) + F(2n-2) would surely fail
Finally the Fibonacci sequence can be explicitly solved with Binet's formula
F_n = int(( (1 + sqrt5) ** n - (1 - sqrt5) ** n ) / ( 2 ** n * sqrt5))
How would a test be performed to find out if it is feasible to solve such a sequence, and if so what method from first principles would be applied in order to solve?
I'm particularly interested in how such solutions are derived.
Thank you very much!