Counting Bits with Fibonacci and Zeckendorf
نویسندگان
چکیده
Fibonacci numbers and difference equations show up in many counting problems. Zeckendorf showed how to represent natural numbers in “binary” Fibonacci bases. Capocelli counted the number of 0 bits and 1 bits in such representation. Here we use the theory of difference equations to try to provide proofs for Capocelli’s claims. We also investigate generalization of the Fibonacci difference equation which may show behavior similar to that observed by Capocelli. In particular, we conjecture that “doubly non-negative” difference equations will have solutions whose ratios monotonically approach a limit.
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