A Context-Free Grammar Associated with Fibonacci and Lucas Sequences

We introduce a context-free grammar $G=\left\{s⟶s+d,d⟶s\right\}$ to generate Fibonacci and Lucas sequences. By applying the grammar $G$ , we give a grammatical proof of the Binet formula. Besides, we use the grammar $G$ to provide a unified approach to prove several binomial convolutions about Fibonacci and Lucas numbers, which were given by Hoggatt, Carlitz, and Church. Meanwhile, we also obtain some new binomial convolutions.