A Generalization of Zeckendorf's Theorem Via Circumscribed m-gons

Zeckendorf’s theorem states that every positive integer can be uniquely decom- posed as a sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbers satisfy F1 = 1, F2 = 2, and Fn = Fn−1+Fn−2 for n ≥ 3. The distribution of the number of summands in such a decomposition converges to a Gaussian, the gaps between summands converge to geometric decay, and the distribution of the longest gap is similar to that of the longest run of heads in a biased coin; these results also hold more generally, though for technical reasons previous work is needed to assume the coefficients in the recurrence relation are nonnegative and the first term is positive.

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