Miller ZeckGapsNormalizationConstants HCISSIM 29July2024

Steven Miller: Fibonacci Association: A beautiful theorem of Zeckendorf states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers. Once this has been shown, it's natural to ask how many Fibonacci numbers are needed. Lekkerkerker proved that the average number of such summands needed for integers in [F_n, F_{n+1}) is n / (phi^2 + 1), where phi is the golden mean. We present a combinatorial proof of this through the cookie problem and differentiating identities, and further prove that the fluctuations about the mean are normally distributed and the
distribution of gaps between summands is exponentially decreasing. These
techniques apply to numerous generalizations, which we'll discuss as time