Fibonacci and binomials

Small cases.
- $n=0$: $F_1=1=\binom{0}{0}$.
- $n=1$: $F_2=1=\binom{1}{0}$.
- $n=2$: $F_3=2=\binom{2}{0}+\binom{1}{1}=1+1$.
- $n=3$: $F_4=3=\binom{3}{0}+\binom{2}{1}=1+2=3$.
- $n=4$: $F_5=5=\binom{4}{0}+\binom{3}{1}+\binom{2}{2}=1+3+1=5$.

Pattern. The identity holds for these values.

Conclusion. $F_{n+1}=\sum_{k=0}^{\lfloor n/2\rfloor}\binom{n-k}{k}$.