I am looking into the following puzzle:
Male bees hatch from unfertilized eggs and so have a mother but no father. Female bees hatch from fertilized eggs. How many ancestors does a male have in the $12th$ generation back? How many of these are males?
So I realized that the numbers are following the Fibonacci sequence but noticing the pattern is not a proof (as the question asks for a specific link with the given data).
My attempt of a proof is the following:
Conjecture: the ancestry follows the Fibonnaci series and hence in the $12th$ generation there are $233$ ancestors and $89$ of them are males.
Proof:
Each male bee contributes $1$ (female) ancestor in the ancestry chain while each female bee contributes to $2$ ancestors (male and female).
This means that at each level of the ancestry tree we have by the definition of the problem statement:
$mb_i=fb_{i-1}$
$fb_i=fb_{i-1} + mb_{i-1} = fb_{i-1} + fm_{i -2}$
where $mb$ denotes the number of male bees and $fb$ the number of female bees
The $fb$ number follows the fibonnaci sequence and the $mb$ is the fibonacci number created at the previous level of the ancestry.
The total number of ancestors at generation $i$ is $mb_i + fb_i$ which is a summation of two fibonnaci sequences where one starts at $0$ and the other at $1$. Therefore this leads to the conclusion that the total number is also following the fibonnacci sequence.
I feel that there is a gap in the last $3$ statements of the proof. Is there is something that is omitted but is required in order to consider that we have proven that the total number follows the Fibonnaci?
Is this a correct proof based on the actual structure of the data and not just on observing the numbers produced?
The presentation can be improved, but you have all of the essential elements. I’m going to change your notation, however: I’ll use $f_n$ for the $n$-th Fibonacci number, with $f_0=0$ and $f_1=1$, $m_n$ for the number of male bees $n$ generations back, with $m_0=1$, and $q_n$ for the number of female bees $n$ generations back, with $q_0=0$. As you say
$$m_n=q_{n-1}\tag{1}$$
and
$$q_n=q_{n-1}+m_{n-1}=q_{n-1}+q_{n-2}\,.$$
Moreover, $q_0=0$, and $q_1=q_0+m_0=0+1=1$. This means that the sequence $\langle q_n:n\in\Bbb N\rangle$ follows the same recurrence as the Fibonacci sequence and has the same two initial values, so it must be the same sequence: $q_n=f_n$ for all $n\in\Bbb N$. In particular, $q_{12}=f_{12}=144$. It then follows immediately from $(1)$ that $m_n=f_{n-1}$ for each $n\in\Bbb N$, so $m_{12}=f_{11}=89$.
Finally, as you saw, the total number of ancestors $n$ generations back is
$$q_n+m_n=q_n+q_{n-1}=q_{n+1}=f_{n+1}\,,$$
and in particular the total number of ancestors $12$ generations back is $f_{13}=233$.