Jumping sequences

TL;DR

This paper studies weight-minimizing jump sequences defined on integers, revealing a connection between optimal sequences and Pell numbers when using a specific weight function.

Contribution

It characterizes the structure of weight-minimizing jump sequences and links them to Pell numbers under a particular weight function.

Findings

Optimal jump sequences follow a pattern related to Pell numbers.

The main result characterizes when a number appears in the sequence based on Pell numbers.

The paper introduces a novel connection between jump sequences and classical number theory.

Abstract

An integer sequence a(n) is called a jump sequence if a(1)=1 and 1<=a(n)<n for n>=2. Such a sequence has the property that a^k(n)=a(a(...(a(n))...)) goes to 1 in finitely many steps and we call the pattern (n,a(n),a^2(n),...,a^k(n)=1) a jumping pattern from n down to 1. In this paper we look at jumping sequences which are weight minimizing with respect to various weight functions (where a weight w(i,j) is given to each jump from j down to i). Our main result is to show that if w(i,j)=(i+j)/i^2 then the cost minimizing jump sequence has the property that the number m satisfies m=a^q(p) for arbitrary q and some p (depending on q) if and only if m is a Pell number.