[igotrhythm]

Instead of thinking in terms of the number of mice, try thinking in terms of the number of breeding pairs of mice. Besides making the numbers smaller and thus making it easier to recognize patterns (you mentioned that it was getting tedious doing it step by step), it also leads into discussion of one of my favorite mathematical sequences, which was derived from a very similar problem.

At time 0, there is 1 breeding pair plus 3 immature pairs. The first mature breeding pair kicks out another 3 immature pairs at t=40 and t=80, giving 1 breeding pair plus 9 immature pairs at 80 days.

However, at t=120 days, we have 2 breeding pairs, and mice being the animals they are, they immediately get to work and pop out another 3 immature pairs. So we have 2 mature pairs and 15 immature. At t=160, there's 3 mature and 24 immature, and so on.

Since 1 year is approximately t=360 in this system, we have as follows (white text alert):
Number of mature breeding pairs: 1, 1, 1, 2, 3, 5, 8, 13, 21, 34
Number of immature breeding pairs: 3, 6, 9, 15, 24, 39, 63, 102, 165, 267

If the sequence of numbers of mature breeding pairs for given values of t looks familiar, the pattern should help you with how I derived the number of immature pairs.