Find a recurrence relation for a(n) for the number of bees

Question

Find a recurrence relation for a(n) for the number of bees in the nth previous generation of a male bee, if a male bee is born asexually from a single female and a female bee has the normal male and female parents. Given a(1) = 1, a(2) = 2, a(3) = 3.

Can someone show me how to solve this?

Answer 1

I will assume that each parent is unique, that they sire exactly one offspring each (together in the case of male and female), all mating for a generation coincides and occurs simultaneously according to a set schedule, and they die immediately after.

$a(n)$ is the total number of bees in a generation.

Notice that the number of female bees in a generation is precisely the number of bees in the younger generation since every bee has a mother (possibly a father too but irrelevant for now).

We get then the nice fact that among the $a(n)$ bees, precisely $a(n-1)$ of them are female and the remaining $a(n)-a(n-1)$ are male.

Now, for the next generation, each female bee from the current generation contributes two bees to the total for the next generation while each male from the current generation contributes one bee to the total for the next generation.

We get then $a(n+1) = 2\cdot \underbrace{a(n-1)}_{\#female} + 1\cdot \underbrace{(a(n)-a(n-1))}_{\#male} = a(n)+a(n-1)$, and we recognize now that our sequence is simply the Fibonacci sequence with possible shifting.