TITLE: Is the $k$th Fibonacci's rabbit young or adult?
QUESTION [5 upvotes]: Consider the rabbit Fibonacci problem.
We start with one young rabbit couple.  At each step, young rabbit couple grows to adult rabbits and adult rabbit couples give birth a young couple.  If we denote A for an adult couple and y for a young couple, we do iteratively A$\to$Ay, y$\to$A and the sequence is :
y
A
Ay
AyA
AyAAy
AyAAyAyA
AyAAyAyAAyAAy
AyAAyAyAAyAAyAyAAyAyA
AyAAyAyAAyAAyAyAAyAyAAyAAyAyAAyAAy
AyAAyAyAAyAAyAyAAyAyAAyAAyAyAAyAAyAyAAyAyAAyAAyAyAAyAyA
AyAAyAyAAyAAyAyAAyAyAAyAAyAyAAyAAyAyAAyAyAAyAAyAyAAyAyAAyAAyAyAAyAAyAyAAyAyAAyAAyAyAAyAAy
AyAAyAyAAyAAyAyAAyAyAAyAAyAyAAyAAyAyAAyAyAAyAAyAyAAyAyAAyAAyAyAAyAAyAyAAyAyAAyAAyAyAAyAAyAyAAyAyAAyAAyAyAAyAyAAyAAyAyAAyAAyAyAAyAyAAyAAyAyAAyAyA

Quite funnily, except for the first, y, if the $k$th couple is adult (resp. young), is will stay adult (resp. young) forever (despise the shift due to A$\to$Ay ).
For now, having this fact empirically is enough for me.  My question is : is there a simple way to know whether the $k$th couple of the sequence, if it exists, is an adult couple or a young couple, without iterating back the whole sequence ?
Thank you.

REPLY [3 votes]: This OEIS entry gives the formula*
$$
r_k = \lfloor(k+1)\phi\rfloor-\lfloor k\phi\rfloor - 1
$$
where $\phi = \frac{\sqrt 5 - 1}{2}$ is the golden ratio. To translate into your language, $1$ in the OEIS sequence means A, and $0$ means y. The first A corresponds to $r_1$.
*The formula as given in the OEIS doesn't have the $-1$ term at the end, and yields $1$ and $2$, while the sequence has $0$ and $1$. Alternatively, one could've swapped out $\phi$ with $\phi' = \frac1\phi = \phi-1$ instead of appending $-1$.