Counting Keith numbers
Abstract
A Keith number is a positive integer N with the decimal representation a1a2...an such that n>=2 and N appears in the sequence (Km) given by the recurrence K1=a1,...,Kn=an and Km=Km-1+Km-2+...+Km-n for m>n. We prove that there are only finitely many Keith numbers using only one decimal digit (i.e., a1=a2=...=an), and that the set of Keith numbers is of asymptotic density zero.
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