The Meta-C-finite Ansatz

The Fibonacci numbers satisfy the famous recurrence $F_n = F_{n - 1} + F_{n - 2}$. The theory of C-finite sequences ensures that the Fibonacci numbers whose indices are divisible by $m$, namely $F_{mn}$, satisfy a similar recurrence for every positive integer $m$, and these recurrences have an explicit, uniform representation. We will show that $a(mn)$ has a uniform recurrence over $m$ for any C-finite sequence $a(n)$ and use this to automatically derive some famous summation identities.