Growth of modified binomial recurrence The binomial coefficients $\binom{n}{r}$ satisfies $\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}$. This means it is a solution of the equation $f(n,r)=f(n-1,r)+f(n-1,r-1)$, with initial conditions $f(n,0)=1$ for $n\geq 0$, and $f(0,r)=0$ for $r>0$.
What if we slightly change the recurrence to $f(n,r)=f(n-\textbf{2},r)+f(n-1,r-1)$? Will the asymptotic growth still be the same as binomial coefficients? If not, what will it be? (If initial conditions matter, assume that $f(n,r)=1$ if it cannot be defined by the recurrence.)

If you use the same initial conditions as the binomial coefficients, you get the formula
$$ f(n,r) = \binom{\lfloor \frac{n+r}{2} \rfloor}{r}, $$
from which you can extract asymptotics.
Fun fact: $\sum_{r=0}^n f(n,r) = F_{r+2}$, where $F_r$ is the $r$'th Fibonacci number.
