A generalization of Zeckendorf's theorem via circumscribed m-gons
Abstract
Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a
sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbers satisfy F 1= 1, F
2= 2, and F n= F n− 1+ F n− 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.
sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbers satisfy F 1= 1, F
2= 2, and F n= F n− 1+ F n− 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.
