Generalizing Zeckendorf’s Theorem: The Kentucky Sequence

ABSTRACT. By Zeckendorf’s theorem, an equivalent definition of the Fibonacci sequence (appro-priately normalized) is that it is the unique sequence of increasing integers such that every positive number can be written uniquely as a sum of non-adjacent elements; this is called a legal decomposi-tion. Previous work examined the distribution of the number of summands and the spacings between them, in legal decompositions arising from the Fibonacci numbers and other linear recurrence rela-tions with non-negative integral coefficients. Many of these results were restricted to the case where the first term in the defining recurrence was positive. We study a generalization of the Fibonacci numbers with a simple notion of legality which leads to a recurrence where the first term vanishes. We again have unique legal decompositions, Gaussian behavior in the number of summands, and geometric decay in the distribution of gaps. 1