Jovian hares are hermaphrodite. Each mature Jovian hare produces one leveret during each breeding cycle. Each leveret takes two breeding cycles to mature into a fully-grown hare, and then lives for ever. Starting with a single newly-born Jovian leveret,how many hares (including leverets) are there after 47 breeding cycles?
This is the question that I am being asked to consider. I have found the equation to calculate the number of hares and leverets after $n$ breeding cycles. It is $M_n = M_{n-1}+M_{n-3}$ and have found the number of hares and leverets that will present after 47 breeding cycles. I am just a little iffy on how to write a proof of the recurrence relation. Can someone enlighten me on how to do this for this problem?
How many hares are there after $n$ cycles?
All the hares that were alive last cycle, because they live forever. This is $M_{n-1}$ hares.
One leveret for each fully-grown hare; that is, one for each hare that is at least three cycles old. This is $M_{n-3}$ hares.