> I think the answer to your question is lurking in what I said. Note that I considered a second-order differential equation and a second-order recurrence, but the exact same idea works for the third-order one you are interested in. You just need a \\(3 \times 3\\) matrix instead of a \\(2 \times 2\\).
I believe we are thinking of the same approach.
In another thread I wrote a few recurrence puzzles. I threw a 9th order one in there. I asked folks to compute \\(10^7\\) terms in these recurrences.
The only way I know how to compute so many terms uses your trick...