Let F(n) be the Fibonacci sequence as defined by F(1)=1, F(2)=1, F(n)=F(n-1)+F(n-2) for n>=3. I'm looking for a pure recurrence formula for the function X(i)=F(2^{i}) whose coefficients may be polynomials in i. This is Sloane's A058635. I also would like it to be "pure" in the sense that there is no auxiliary function involved. Is such a formula known?

I attempted using Sister Celine's technique (as described in A=B) with the data up to 2^{21} without success.

My motivation is that I have a fairly complicated recurrence formula for another sequence, but I am only interested in the terms whose indices are of the form 2^{i}-3. The existence (or non-existence) of a recursion for X(i) would be a kind of "proof of concept" as to whether or not I should explore the possibility of finding such a recursion for my sequence.