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.
let
$b_n$ be the number born in the nth month
$y_n$ be the number of pre-fertile youths
$f_n$ the number of fertile pairs
$d_n$ the number of pairs dying
then the total number of pairs is
$$ t_n=b_n+y_n+f_n $$ the recursions are given by :
$b_n=6f_{n-1}$
$y_n=b_{n-1}$
$f_n=f_{n-1}+y_{n-1}-d_n$
$d_n=b_{n-6}$
Which can be simplified to ..
$b_n=6f_{n-1}$
$f_n=f_{n-1}+b_{n-2}-b_{n-6}$
So $$t_n=b_n+b_{n-1} +f_n \\t_n=6f_{n-1}+6f_{n-2} + f_n $$ and $$f_n = f_{n-1} + 6f_{n-3} -6f_{n-7} $$
So you have the correct recurrence for the number of fertile pairs but not the total number of living pairs
This is what it looks like for the first 10 months ... $$\begin{array} \\n & b_n & y_n & f_n & d_n \\0 &1 &0 & 0 & 0 \\1 &0 &1 & 0 &0 \\2& 0 &0 &1 & 0 \\3 &6 &0 & 1 &0 \\4 &6 &6 & 1 & 0 \\5 &6 &6 & 7 & 0 \\6 &42 &6 & 13 & 0 \\7 &78 &42 & 18 & 1 \\8 &108 &78 & 60 & 0 \\9 &360 &108 & 138 & 0 \\10 &828 &360 & 240 & 6 \end{array} $$