I came across the following question on a math test:
Suppose Fibonacci's research in the breeding habits of rabbits has been adjusted. They are now believed to be fertile after $2$ months of life, and they consistently give birth $6$ pairs of rabbits at the end of every month; the gestation period is still one month (it takes $1$ month for them to give birth after they are fertile). Also, these particular rabbits only live $6$ months.
Find a recurrence relation for $s_n$
For this problem I first started to list the total number of rabbits after $n$ number of months:
$$s_0 = 1$$ $$s_1 = 1$$ $$s_2 = 1$$ $$s_3 = 1 + 6 = 7$$ $$s_4 = 7 + 6 = 13$$ $$s_5 = 19$$ $$s_6 = 61$$ $$s_7 = 133$$ $$\vdots$$
After using this recurrence pattern, I got the following sequence:
$$s_n = s_{n-1} + 6s_{n-3} - 6s_{n-7}$$
However, my teacher told me the answer was incorrect.
I have no idea where my math went wrong. Any help would be appreciated.