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

Description

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.