I was trying to simplify the following sequence, such that I can calculate the $n$th term in $\log n$ time. This can be done, if we can express the $n$th in terms of combinations of Fibonacci like numbers.
$S_{n} = (S_{n-1} * S_{n-2} )^k$ given that $S_{0} = a$, $S_{1} = b$ and $k$ is an fixed integer.
I tried my best but could not find a simple formula. Is it possible to get some simple generating function so that I can convert that to Matrix form for calculating in $S_{n}$ in $\log n$ time.