As we know for all integers $k\geq{2}$, $F_k = F_k{_-}{_1} + F_k{_-}{_2}$
So the first few numbers of the Fibonacci sequence are 0,1,1,2,3,5,8, etc.
I'm studying discrete mathematics right now by Epp. I'm learning about recursive sequences right now, and am confused by the wording of the conditions that is proposed by Fibonacci.
A single pair of rabbits (male and female) is born at the beginning of a year. Assume the following conditions: (1) Rabbit pairs are not fertile during their first month of life, but thereafter give birth to one new male/female pairs at the end of every month. (2) No rabbits die.
Here is what I make of it. I must be misinterpreting the wording and would like it for someone to point out my mistake in my reasoning below. Any help is greatly appreciated.
You start out with one rabbit-pair. The conditions say that this rabbit is not fertile during their first month of life. The rabbit pair is born at the beginning of a year, so they are born January 1st. There are 30/31 in a month (excluding a leap year month). My reasoning is that their first month of life would be from January 1st, to February 1st. They are not fertile during these 30/31 days, so at the end of January, their is only 1 rabbit pair, themselves.
Let $F_n$ = The number of rabbit-pairs alive at end of month n.
So $F_0 = 1$
Now that they have LIVED for a month (their first month of life), they should be fertile now and at the end of February another rabbit pair should be born.
So $F_1 = 2$
However it says in the textbook "Now $F_0 = 1$, as already noted, and $F_1 = 1$ also, because the first pair of rabbits is not fertile until the second month."
So, does "not fertile during their first month of life" mean that instead their first 30 days of life are considered their "zeroth month of life" and the month following that is their "first month of life"? It would only be after this, the initial rabbit pair gives birth to a new rabbit pair at the end of the second month. Is this correct?