Fibonacci Identities Derived from Path Counting in Automata
نویسندگان
چکیده
Combinatorial proofs are appealing since they lead to intuitive understanding. Proofs based on other mathematical techniques may be convincing, but still leave the reader wondering why the result holds. A large collection of combinatorial proofs is presented in [1], including many proofs of Fibonacci identities based on counting tilings of a one-dimensional board with squares and dominoes. An alternative approach, much in the same spirit, is to base proofs on automata that recognize, i.e., accept, such tilings. Although many of the same results can be obtained, we will show here that the automata-based approach in some cases has interesting advantages. As a simple example, we may start with a deterministic finite state automaton in Figure 1, which corresponds very directly to square-domino tilings.
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