Abstract
We introduce a context-free grammar G = s ⟶ s + d , d ⟶ s 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.
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