I was intrigued by a question asked by another user, and although I didn't provide an answer, I engaged in externded chat with a user who did. He gave me lots of great advice, and gave me a lot of his time. The last comment ended in him pointing me in the right direction. I don't feel I can ask any more of him, since he has already helped me in understanding the problem so much already. Is it acceptable to pose a question based on my partital understanding of the problem, referencing his ideas and requesting clarification?
To clarify, a link to the question (and the numerous comments!) can be found here.
The comment I require clarification (and possibly extended explanation of) is:
that limit is the ratio of the coefficients of the eigenvector corresponding to the larger eigenvalue (which you can take at any $n≥216$ because it doesn't change beyond $216$). So compute ($a_{216},b_{216}$) for both strategies, decompose that with respect to the eigensystem of the recurrence matrix, and form the ratio of the coefficients corresponding to the larger eigenvalue.
I was not unfortunately able to achieve this.