Can Fibonacci Sequences Be Proven Using This Conjecture?

[ramsey2879]

Given $$A_1 = 3 \texttt{ and } A_n = A_{n-1}^{2} -2$$; is there a way to prove the following:

$$\prod_{i=1}^{n}A_{i} = F_{2^{n+1}}$$

or if someone has already proven this, can you give the reference?

 

[ramsey2879]

Note of is relation can be found at the first comment re the sequence 3,7,47 ... See http://www.research.att.com/~njas/sequences/A001566" , but I would like a proof if it is known.

 

[MrJB]

These two identities should get you there.

$$F_{2n}=L_{n}F_{n}$$

$$L_{2n}=L_{n}^{2}-2(-1)^{n}$$

Where $$L_n$$ is the nth Lucas numbers.

 

[ramsey2879]

These two identities should get you there.

$$F_{2n}=L_{n}F_{n}$$

$$L_{2n}=L_{n}^{2}-2(-1)^{n}$$

Where $$L_n$$ is the nth Lucas numbers.

Where can I find a proof of those identities?

 

[MrJB]

Both identities can be proven using the closed form expressions for the Fibonacci and Lucas numbers. The closed form expressions should be easily accessible. The one for the Fibonacci numbers is also known as Binet's formula.