Abstract

Three equivalent methods of generating the paperfolding sequence are presented as well as a categorization of runs of identical terms. We find all repeated subsequences, the largest repeated subsequences and the spacing of singles, doubles and triples throughout the sequence. The paperfolding sequence is shown to have links to the Binary Reflected Gray Code and the Stern-Brocot tree.

Highlights

  • Take a sheet of paper and fold it, right over left, n times

  • The middle element of Sn is always a 1; Sn−1 appears to the left of this 1 and SnR−1 appears to the right

  • SOME FUNCTIONS RELATED TO THE PAPERFOLDING SEQUENCE

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Summary

INTRODUCTION

Take a sheet of paper and fold it, right over left, n times. When the paper is unfolded we see a sequence of 2n −1 creases, some downward and some upward. Prodinger and Urbanek [14] label them 0 and 1 while Allouche and Bousquet-Melou [4] allow both 1 and 0 and 0 and 1 This sequence of 2n − 1 1s and 0s we call Sn as do Davis and Knuth [1] with their sequence of 2n − 1 Ds and U s. Mendes France and Shallit [12] give four different methods for representing the sequence One of their representations, called the Dragon Curve in Davis and Knuth [1], is a sequence of lattice points obtained by unfolding the paper so that all the folds are 900 and looking at the edge of the paper. Davis and Knuth [1] and Prodinger and Urbanek [14] have yet another method for constructing Sn and S This can be expressed through interleaving, as follows. We link the Paperfolding and Stern-Brocot sequences, examine functions related to S and show that one of these has properties similar to those of the Gray code function of Bunder et al [7]

THE GENERAL ALTERNATING PAPERFOLDING SEQUENCE
SOME FUNCTIONS RELATED TO THE PAPERFOLDING SEQUENCE
REPEATED SUBSEQUENCS OF THE PAPERFOLDING SEQUENCE
POWER SERIES AND THE PAPERFOLDING SEQUENCE
LINKING PAPERFOLDING AND THE STERN-BROCOT TREE
LINKING PAPERFOLDING AND THE BINARY REFLECTED GRAY CODE
NUMBERED PAPERFOLDING SEQUENCE
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