paper

On Curling Numbers of Integer Sequences

arXiv:1212.6102

Abstract

Given a finite nonempty sequence S of integers, write it as XY^k, where Y^k is a power of greatest exponent that is a suffix of S: this k is the curling number of S. The Curling Number Conjecture is that if one starts with any initial sequence S, and extends it by repeatedly appending the curling number of the current sequence, the sequence will eventually reach 1. The conjecture remains open. In this paper we discuss the special case when S consists just of 2's and 3's. Even this case remains open, but we determine how far a sequence of n 2's and 3's can extend before reaching a 1, conjecturally for n <= 80. We investigate several related combinatorial problems, such as finding c(n,k), the number of binary sequences of length n and curling number k, and t(n,i), the number of sequences of length n which extend for i steps before reaching a 1. A number of interesting combinatorial problems remain unsolved.

25 pages, one figure, 14 tables. This paper is a sequel to the paper arXiv:0912.2382. Feb 17 2013: added list of OEIS sequences that are mentioned. March 12 2013: A number of small improvements

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